Theorem gruina 9640

Description: If a Grothendieck universe U is nonempty, then the height of the ordinals in U is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.)

Hypothesis

Ref	Expression
gruina.1	|- A = ( U i^i On )

Assertion

Ref	Expression
gruina	|- ( ( U e. Univ /\ U =/= (/) ) -> A e. Inacc )

Proof of Theorem gruina

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . . 4 |- ( U =/= (/) <-> E. x x e. U )
2		0ss 3972	. . . . . . . . . 10 |- (/) C_ x
3		gruss 9618	. . . . . . . . . 10 |- ( ( U e. Univ /\ x e. U /\ (/) C_ x ) -> (/) e. U )
4	2, 3	mp3an3 1413	. . . . . . . . 9 |- ( ( U e. Univ /\ x e. U ) -> (/) e. U )
5		0elon 5778	. . . . . . . . 9 |- (/) e. On
6		elin 3796	. . . . . . . . 9 |- ( (/) e. ( U i^i On ) <-> ( (/) e. U /\ (/) e. On ) )
7	4, 5, 6	sylanblrc 697	. . . . . . . 8 |- ( ( U e. Univ /\ x e. U ) -> (/) e. ( U i^i On ) )
8		gruina.1	. . . . . . . 8 |- A = ( U i^i On )
9	7, 8	syl6eleqr 2712	. . . . . . 7 |- ( ( U e. Univ /\ x e. U ) -> (/) e. A )
10		ne0i 3921	. . . . . . 7 |- ( (/) e. A -> A =/= (/) )
11	9, 10	syl 17	. . . . . 6 |- ( ( U e. Univ /\ x e. U ) -> A =/= (/) )
12	11	expcom 451	. . . . 5 |- ( x e. U -> ( U e. Univ -> A =/= (/) ) )
13	12	exlimiv 1858	. . . 4 |- ( E. x x e. U -> ( U e. Univ -> A =/= (/) ) )
14	1, 13	sylbi 207	. . 3 |- ( U =/= (/) -> ( U e. Univ -> A =/= (/) ) )
15	14	impcom 446	. 2 |- ( ( U e. Univ /\ U =/= (/) ) -> A =/= (/) )
16		grutr 9615	. . . . . . . 8 |- ( U e. Univ -> Tr U )
17		tron 5746	. . . . . . . 8 |- Tr On
18		trin 4763	. . . . . . . 8 |- ( ( Tr U /\ Tr On ) -> Tr ( U i^i On ) )
19	16, 17, 18	sylancl 694	. . . . . . 7 |- ( U e. Univ -> Tr ( U i^i On ) )
20		inss2 3834	. . . . . . . 8 |- ( U i^i On ) C_ On
21		epweon 6983	. . . . . . . 8 |- _E We On
22		wess 5101	. . . . . . . 8 |- ( ( U i^i On ) C_ On -> ( _E We On -> _E We ( U i^i On ) ) )
23	20, 21, 22	mp2 9	. . . . . . 7 |- _E We ( U i^i On )
24		df-ord 5726	. . . . . . 7 |- ( Ord ( U i^i On ) <-> ( Tr ( U i^i On ) /\ _E We ( U i^i On ) ) )
25	19, 23, 24	sylanblrc 697	. . . . . 6 |- ( U e. Univ -> Ord ( U i^i On ) )
26		inex1g 4801	. . . . . 6 |- ( U e. Univ -> ( U i^i On ) e. _V )
27		elon2 5734	. . . . . 6 |- ( ( U i^i On ) e. On <-> ( Ord ( U i^i On ) /\ ( U i^i On ) e. _V ) )
28	25, 26, 27	sylanbrc 698	. . . . 5 |- ( U e. Univ -> ( U i^i On ) e. On )
29	8, 28	syl5eqel 2705	. . . 4 |- ( U e. Univ -> A e. On )
30	29	adantr 481	. . 3 |- ( ( U e. Univ /\ U =/= (/) ) -> A e. On )
31		eloni 5733	. . . . . . 7 |- ( A e. On -> Ord A )
32		ordirr 5741	. . . . . . 7 |- ( Ord A -> -. A e. A )
33	31, 32	syl 17	. . . . . 6 |- ( A e. On -> -. A e. A )
34		elin 3796	. . . . . . . . 9 |- ( A e. ( U i^i On ) <-> ( A e. U /\ A e. On ) )
35	34	biimpri 218	. . . . . . . 8 |- ( ( A e. U /\ A e. On ) -> A e. ( U i^i On ) )
36	35, 8	syl6eleqr 2712	. . . . . . 7 |- ( ( A e. U /\ A e. On ) -> A e. A )
37	36	expcom 451	. . . . . 6 |- ( A e. On -> ( A e. U -> A e. A ) )
38	33, 37	mtod 189	. . . . 5 |- ( A e. On -> -. A e. U )
39	30, 38	syl 17	. . . 4 |- ( ( U e. Univ /\ U =/= (/) ) -> -. A e. U )
40		inss1 3833	. . . . . . . . . . . . . . . 16 |- ( U i^i On ) C_ U
41	8, 40	eqsstri 3635	. . . . . . . . . . . . . . 15 |- A C_ U
42	41	sseli 3599	. . . . . . . . . . . . . 14 |- ( x e. A -> x e. U )
43		vpwex 4849	. . . . . . . . . . . . . . . 16 |- ~P x e. _V
44	43	canth2 8113	. . . . . . . . . . . . . . 15 |- ~P x ~< ~P ~P x
45	43	pwex 4848	. . . . . . . . . . . . . . . . . 18 |- ~P ~P x e. _V
46	45	cardid 9369	. . . . . . . . . . . . . . . . 17 |- ( card ` ~P ~P x ) ~~ ~P ~P x
47	46	ensymi 8006	. . . . . . . . . . . . . . . 16 |- ~P ~P x ~~ ( card ` ~P ~P x )
48	29	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( U e. Univ /\ x e. U ) -> A e. On )
49		grupw 9617	. . . . . . . . . . . . . . . . . . 19 |- ( ( U e. Univ /\ x e. U ) -> ~P x e. U )
50		grupw 9617	. . . . . . . . . . . . . . . . . . 19 |- ( ( U e. Univ /\ ~P x e. U ) -> ~P ~P x e. U )
51	49, 50	syldan 487	. . . . . . . . . . . . . . . . . 18 |- ( ( U e. Univ /\ x e. U ) -> ~P ~P x e. U )
52	29	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( U e. Univ /\ ~P ~P x e. U ) -> A e. On )
53		endom 7982	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( card ` ~P ~P x ) ~~ ~P ~P x -> ( card ` ~P ~P x ) ~<_ ~P ~P x )
54	46, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 |- ( card ` ~P ~P x ) ~<_ ~P ~P x
55		cardon 8770	. . . . . . . . . . . . . . . . . . . . . 22 |- ( card ` ~P ~P x ) e. On
56		grudomon 9639	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( U e. Univ /\ ( card ` ~P ~P x ) e. On /\ ( ~P ~P x e. U /\ ( card ` ~P ~P x ) ~<_ ~P ~P x ) ) -> ( card ` ~P ~P x ) e. U )
57	55, 56	mp3an2 1412	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( U e. Univ /\ ( ~P ~P x e. U /\ ( card ` ~P ~P x ) ~<_ ~P ~P x ) ) -> ( card ` ~P ~P x ) e. U )
58	54, 57	mpanr2 720	. . . . . . . . . . . . . . . . . . . 20 |- ( ( U e. Univ /\ ~P ~P x e. U ) -> ( card ` ~P ~P x ) e. U )
59		elin 3796	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( card ` ~P ~P x ) e. ( U i^i On ) <-> ( ( card ` ~P ~P x ) e. U /\ ( card ` ~P ~P x ) e. On ) )
60	59	biimpri 218	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( card ` ~P ~P x ) e. U /\ ( card ` ~P ~P x ) e. On ) -> ( card ` ~P ~P x ) e. ( U i^i On ) )
61	60, 8	syl6eleqr 2712	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( card ` ~P ~P x ) e. U /\ ( card ` ~P ~P x ) e. On ) -> ( card ` ~P ~P x ) e. A )
62	58, 55, 61	sylancl 694	. . . . . . . . . . . . . . . . . . 19 |- ( ( U e. Univ /\ ~P ~P x e. U ) -> ( card ` ~P ~P x ) e. A )
63		onelss 5766	. . . . . . . . . . . . . . . . . . 19 |- ( A e. On -> ( ( card ` ~P ~P x ) e. A -> ( card ` ~P ~P x ) C_ A ) )
64	52, 62, 63	sylc 65	. . . . . . . . . . . . . . . . . 18 |- ( ( U e. Univ /\ ~P ~P x e. U ) -> ( card ` ~P ~P x ) C_ A )
65	51, 64	syldan 487	. . . . . . . . . . . . . . . . 17 |- ( ( U e. Univ /\ x e. U ) -> ( card ` ~P ~P x ) C_ A )
66		ssdomg 8001	. . . . . . . . . . . . . . . . 17 |- ( A e. On -> ( ( card ` ~P ~P x ) C_ A -> ( card ` ~P ~P x ) ~<_ A ) )
67	48, 65, 66	sylc 65	. . . . . . . . . . . . . . . 16 |- ( ( U e. Univ /\ x e. U ) -> ( card ` ~P ~P x ) ~<_ A )
68		endomtr 8014	. . . . . . . . . . . . . . . 16 |- ( ( ~P ~P x ~~ ( card ` ~P ~P x ) /\ ( card ` ~P ~P x ) ~<_ A ) -> ~P ~P x ~<_ A )
69	47, 67, 68	sylancr 695	. . . . . . . . . . . . . . 15 |- ( ( U e. Univ /\ x e. U ) -> ~P ~P x ~<_ A )
70		sdomdomtr 8093	. . . . . . . . . . . . . . 15 |- ( ( ~P x ~< ~P ~P x /\ ~P ~P x ~<_ A ) -> ~P x ~< A )
71	44, 69, 70	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( U e. Univ /\ x e. U ) -> ~P x ~< A )
72	42, 71	sylan2 491	. . . . . . . . . . . . 13 |- ( ( U e. Univ /\ x e. A ) -> ~P x ~< A )
73	72	ralrimiva 2966	. . . . . . . . . . . 12 |- ( U e. Univ -> A. x e. A ~P x ~< A )
74		inawinalem 9511	. . . . . . . . . . . 12 |- ( A e. On -> ( A. x e. A ~P x ~< A -> A. x e. A E. y e. A x ~< y ) )
75	29, 73, 74	sylc 65	. . . . . . . . . . 11 |- ( U e. Univ -> A. x e. A E. y e. A x ~< y )
76	75	adantr 481	. . . . . . . . . 10 |- ( ( U e. Univ /\ U =/= (/) ) -> A. x e. A E. y e. A x ~< y )
77		winainflem 9515	. . . . . . . . . 10 |- ( ( A =/= (/) /\ A e. On /\ A. x e. A E. y e. A x ~< y ) -> om C_ A )
78	15, 30, 76, 77	syl3anc 1326	. . . . . . . . 9 |- ( ( U e. Univ /\ U =/= (/) ) -> om C_ A )
79		vex 3203	. . . . . . . . . . . . . . 15 |- x e. _V
80	79	canth2 8113	. . . . . . . . . . . . . 14 |- x ~< ~P x
81		sdomtr 8098	. . . . . . . . . . . . . 14 |- ( ( x ~< ~P x /\ ~P x ~< A ) -> x ~< A )
82	80, 72, 81	sylancr 695	. . . . . . . . . . . . 13 |- ( ( U e. Univ /\ x e. A ) -> x ~< A )
83	82	ralrimiva 2966	. . . . . . . . . . . 12 |- ( U e. Univ -> A. x e. A x ~< A )
84		iscard 8801	. . . . . . . . . . . 12 |- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. A x ~< A ) )
85	29, 83, 84	sylanbrc 698	. . . . . . . . . . 11 |- ( U e. Univ -> ( card ` A ) = A )
86		cardlim 8798	. . . . . . . . . . . 12 |- ( om C_ ( card ` A ) <-> Lim ( card ` A ) )
87		sseq2 3627	. . . . . . . . . . . . 13 |- ( ( card ` A ) = A -> ( om C_ ( card ` A ) <-> om C_ A ) )
88		limeq 5735	. . . . . . . . . . . . 13 |- ( ( card ` A ) = A -> ( Lim ( card ` A ) <-> Lim A ) )
89	87, 88	bibi12d 335	. . . . . . . . . . . 12 |- ( ( card ` A ) = A -> ( ( om C_ ( card ` A ) <-> Lim ( card ` A ) ) <-> ( om C_ A <-> Lim A ) ) )
90	86, 89	mpbii 223	. . . . . . . . . . 11 |- ( ( card ` A ) = A -> ( om C_ A <-> Lim A ) )
91	85, 90	syl 17	. . . . . . . . . 10 |- ( U e. Univ -> ( om C_ A <-> Lim A ) )
92	91	adantr 481	. . . . . . . . 9 |- ( ( U e. Univ /\ U =/= (/) ) -> ( om C_ A <-> Lim A ) )
93	78, 92	mpbid 222	. . . . . . . 8 |- ( ( U e. Univ /\ U =/= (/) ) -> Lim A )
94		cflm 9072	. . . . . . . 8 |- ( ( A e. On /\ Lim A ) -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } )
95	30, 93, 94	syl2anc 693	. . . . . . 7 |- ( ( U e. Univ /\ U =/= (/) ) -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } )
96		cardon 8770	. . . . . . . . . . . 12 |- ( card ` y ) e. On
97		eleq1 2689	. . . . . . . . . . . 12 |- ( x = ( card ` y ) -> ( x e. On <-> ( card ` y ) e. On ) )
98	96, 97	mpbiri 248	. . . . . . . . . . 11 |- ( x = ( card ` y ) -> x e. On )
99	98	adantr 481	. . . . . . . . . 10 |- ( ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) -> x e. On )
100	99	exlimiv 1858	. . . . . . . . 9 |- ( E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) -> x e. On )
101	100	abssi 3677	. . . . . . . 8 |- { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } C_ On
102		fvex 6201	. . . . . . . . . 10 |- ( cf ` A ) e. _V
103	95, 102	syl6eqelr 2710	. . . . . . . . 9 |- ( ( U e. Univ /\ U =/= (/) ) -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } e. _V )
104		intex 4820	. . . . . . . . 9 |- ( { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } =/= (/) <-> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } e. _V )
105	103, 104	sylibr 224	. . . . . . . 8 |- ( ( U e. Univ /\ U =/= (/) ) -> { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } =/= (/) )
106		onint 6995	. . . . . . . 8 |- ( ( { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } C_ On /\ { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } =/= (/) ) -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } e. { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } )
107	101, 105, 106	sylancr 695	. . . . . . 7 |- ( ( U e. Univ /\ U =/= (/) ) -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } e. { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } )
108	95, 107	eqeltrd 2701	. . . . . 6 |- ( ( U e. Univ /\ U =/= (/) ) -> ( cf ` A ) e. { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } )
109		eqeq1 2626	. . . . . . . . 9 |- ( x = ( cf ` A ) -> ( x = ( card ` y ) <-> ( cf ` A ) = ( card ` y ) ) )
110	109	anbi1d 741	. . . . . . . 8 |- ( x = ( cf ` A ) -> ( ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) <-> ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) ) )
111	110	exbidv 1850	. . . . . . 7 |- ( x = ( cf ` A ) -> ( E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) <-> E. y ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) ) )
112	102, 111	elab 3350	. . . . . 6 |- ( ( cf ` A ) e. { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) } <-> E. y ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) )
113	108, 112	sylib 208	. . . . 5 |- ( ( U e. Univ /\ U =/= (/) ) -> E. y ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) )
114		simp2rr 1131	. . . . . . . 8 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> A = U. y )
115		simp1l 1085	. . . . . . . . 9 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> U e. Univ )
116		simp2rl 1130	. . . . . . . . . . 11 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> y C_ A )
117	116, 41	syl6ss 3615	. . . . . . . . . 10 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> y C_ U )
118	41	sseli 3599	. . . . . . . . . . 11 |- ( ( cf ` A ) e. A -> ( cf ` A ) e. U )
119	118	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> ( cf ` A ) e. U )
120		simp2l 1087	. . . . . . . . . . 11 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> ( cf ` A ) = ( card ` y ) )
121		vex 3203	. . . . . . . . . . . 12 |- y e. _V
122	121	cardid 9369	. . . . . . . . . . 11 |- ( card ` y ) ~~ y
123	120, 122	syl6eqbr 4692	. . . . . . . . . 10 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> ( cf ` A ) ~~ y )
124		gruen 9634	. . . . . . . . . 10 |- ( ( U e. Univ /\ y C_ U /\ ( ( cf ` A ) e. U /\ ( cf ` A ) ~~ y ) ) -> y e. U )
125	115, 117, 119, 123, 124	syl112anc 1330	. . . . . . . . 9 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> y e. U )
126		gruuni 9622	. . . . . . . . 9 |- ( ( U e. Univ /\ y e. U ) -> U. y e. U )
127	115, 125, 126	syl2anc 693	. . . . . . . 8 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> U. y e. U )
128	114, 127	eqeltrd 2701	. . . . . . 7 |- ( ( ( U e. Univ /\ U =/= (/) ) /\ ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) /\ ( cf ` A ) e. A ) -> A e. U )
129	128	3exp 1264	. . . . . 6 |- ( ( U e. Univ /\ U =/= (/) ) -> ( ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) -> ( ( cf ` A ) e. A -> A e. U ) ) )
130	129	exlimdv 1861	. . . . 5 |- ( ( U e. Univ /\ U =/= (/) ) -> ( E. y ( ( cf ` A ) = ( card ` y ) /\ ( y C_ A /\ A = U. y ) ) -> ( ( cf ` A ) e. A -> A e. U ) ) )
131	113, 130	mpd 15	. . . 4 |- ( ( U e. Univ /\ U =/= (/) ) -> ( ( cf ` A ) e. A -> A e. U ) )
132	39, 131	mtod 189	. . 3 |- ( ( U e. Univ /\ U =/= (/) ) -> -. ( cf ` A ) e. A )
133		cfon 9077	. . . . 5 |- ( cf ` A ) e. On
134		cfle 9076	. . . . . 6 |- ( cf ` A ) C_ A
135		onsseleq 5765	. . . . . 6 |- ( ( ( cf ` A ) e. On /\ A e. On ) -> ( ( cf ` A ) C_ A <-> ( ( cf ` A ) e. A \/ ( cf ` A ) = A ) ) )
136	134, 135	mpbii 223	. . . . 5 |- ( ( ( cf ` A ) e. On /\ A e. On ) -> ( ( cf ` A ) e. A \/ ( cf ` A ) = A ) )
137	133, 136	mpan 706	. . . 4 |- ( A e. On -> ( ( cf ` A ) e. A \/ ( cf ` A ) = A ) )
138	137	ord 392	. . 3 |- ( A e. On -> ( -. ( cf ` A ) e. A -> ( cf ` A ) = A ) )
139	30, 132, 138	sylc 65	. 2 |- ( ( U e. Univ /\ U =/= (/) ) -> ( cf ` A ) = A )
140	73	adantr 481	. 2 |- ( ( U e. Univ /\ U =/= (/) ) -> A. x e. A ~P x ~< A )
141		elina 9509	. 2 |- ( A e. Inacc <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) )
142	15, 139, 140, 141	syl3anbrc 1246	1 |- ( ( U e. Univ /\ U =/= (/) ) -> A e. Inacc )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   class class class wbr 4653   Tr wtr 4752   _E cep 5028   We wwe 5072   Ord word 5722   Oncon0 5723   Lim wlim 5724   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   cfccf 8763   Inacccina 9505   Univcgru 9612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765  df-cf 8767  df-ac 8939  df-ina 9507  df-gru 9613

This theorem is referenced by:  grur1a  9641  grur1  9642  grutsk  9644