Theorem r1tr 8639

Description: The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
r1tr	|- Tr ( R1 ` A )

Proof of Theorem r1tr

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

Step	Hyp	Ref	Expression
1		r1funlim 8629	. . . . . 6 |- ( Fun R1 /\ Lim dom R1 )
2	1	simpri 478	. . . . 5 |- Lim dom R1
3		limord 5784	. . . . 5 |- ( Lim dom R1 -> Ord dom R1 )
4		ordsson 6989	. . . . 5 |- ( Ord dom R1 -> dom R1 C_ On )
5	2, 3, 4	mp2b 10	. . . 4 |- dom R1 C_ On
6	5	sseli 3599	. . 3 |- ( A e. dom R1 -> A e. On )
7		fveq2 6191	. . . . . 6 |- ( x = (/) -> ( R1 ` x ) = ( R1 ` (/) ) )
8		r10 8631	. . . . . 6 |- ( R1 ` (/) ) = (/)
9	7, 8	syl6eq 2672	. . . . 5 |- ( x = (/) -> ( R1 ` x ) = (/) )
10		treq 4758	. . . . 5 |- ( ( R1 ` x ) = (/) -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
11	9, 10	syl 17	. . . 4 |- ( x = (/) -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
12		fveq2 6191	. . . . 5 |- ( x = y -> ( R1 ` x ) = ( R1 ` y ) )
13		treq 4758	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` y ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` y ) ) )
14	12, 13	syl 17	. . . 4 |- ( x = y -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` y ) ) )
15		fveq2 6191	. . . . 5 |- ( x = suc y -> ( R1 ` x ) = ( R1 ` suc y ) )
16		treq 4758	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` suc y ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` suc y ) ) )
17	15, 16	syl 17	. . . 4 |- ( x = suc y -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` suc y ) ) )
18		fveq2 6191	. . . . 5 |- ( x = A -> ( R1 ` x ) = ( R1 ` A ) )
19		treq 4758	. . . . 5 |- ( ( R1 ` x ) = ( R1 ` A ) -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` A ) ) )
20	18, 19	syl 17	. . . 4 |- ( x = A -> ( Tr ( R1 ` x ) <-> Tr ( R1 ` A ) ) )
21		tr0 4764	. . . 4 |- Tr (/)
22		limsuc 7049	. . . . . . . 8 |- ( Lim dom R1 -> ( y e. dom R1 <-> suc y e. dom R1 ) )
23	2, 22	ax-mp 5	. . . . . . 7 |- ( y e. dom R1 <-> suc y e. dom R1 )
24		simpr 477	. . . . . . . . 9 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ( R1 ` y ) )
25		pwtr 4921	. . . . . . . . 9 |- ( Tr ( R1 ` y ) <-> Tr ~P ( R1 ` y ) )
26	24, 25	sylib 208	. . . . . . . 8 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ~P ( R1 ` y ) )
27		r1sucg 8632	. . . . . . . . 9 |- ( y e. dom R1 -> ( R1 ` suc y ) = ~P ( R1 ` y ) )
28		treq 4758	. . . . . . . . 9 |- ( ( R1 ` suc y ) = ~P ( R1 ` y ) -> ( Tr ( R1 ` suc y ) <-> Tr ~P ( R1 ` y ) ) )
29	27, 28	syl 17	. . . . . . . 8 |- ( y e. dom R1 -> ( Tr ( R1 ` suc y ) <-> Tr ~P ( R1 ` y ) ) )
30	26, 29	syl5ibrcom 237	. . . . . . 7 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> ( y e. dom R1 -> Tr ( R1 ` suc y ) ) )
31	23, 30	syl5bir 233	. . . . . 6 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> ( suc y e. dom R1 -> Tr ( R1 ` suc y ) ) )
32		ndmfv 6218	. . . . . . . 8 |- ( -. suc y e. dom R1 -> ( R1 ` suc y ) = (/) )
33		treq 4758	. . . . . . . 8 |- ( ( R1 ` suc y ) = (/) -> ( Tr ( R1 ` suc y ) <-> Tr (/) ) )
34	32, 33	syl 17	. . . . . . 7 |- ( -. suc y e. dom R1 -> ( Tr ( R1 ` suc y ) <-> Tr (/) ) )
35	21, 34	mpbiri 248	. . . . . 6 |- ( -. suc y e. dom R1 -> Tr ( R1 ` suc y ) )
36	31, 35	pm2.61d1 171	. . . . 5 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ( R1 ` suc y ) )
37	36	ex 450	. . . 4 |- ( y e. On -> ( Tr ( R1 ` y ) -> Tr ( R1 ` suc y ) ) )
38		triun 4766	. . . . . . . 8 |- ( A. y e. x Tr ( R1 ` y ) -> Tr U_ y e. x ( R1 ` y ) )
39		r1limg 8634	. . . . . . . . . 10 |- ( ( x e. dom R1 /\ Lim x ) -> ( R1 ` x ) = U_ y e. x ( R1 ` y ) )
40	39	ancoms 469	. . . . . . . . 9 |- ( ( Lim x /\ x e. dom R1 ) -> ( R1 ` x ) = U_ y e. x ( R1 ` y ) )
41		treq 4758	. . . . . . . . 9 |- ( ( R1 ` x ) = U_ y e. x ( R1 ` y ) -> ( Tr ( R1 ` x ) <-> Tr U_ y e. x ( R1 ` y ) ) )
42	40, 41	syl 17	. . . . . . . 8 |- ( ( Lim x /\ x e. dom R1 ) -> ( Tr ( R1 ` x ) <-> Tr U_ y e. x ( R1 ` y ) ) )
43	38, 42	syl5ibr 236	. . . . . . 7 |- ( ( Lim x /\ x e. dom R1 ) -> ( A. y e. x Tr ( R1 ` y ) -> Tr ( R1 ` x ) ) )
44	43	impancom 456	. . . . . 6 |- ( ( Lim x /\ A. y e. x Tr ( R1 ` y ) ) -> ( x e. dom R1 -> Tr ( R1 ` x ) ) )
45		ndmfv 6218	. . . . . . . 8 |- ( -. x e. dom R1 -> ( R1 ` x ) = (/) )
46	45, 10	syl 17	. . . . . . 7 |- ( -. x e. dom R1 -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
47	21, 46	mpbiri 248	. . . . . 6 |- ( -. x e. dom R1 -> Tr ( R1 ` x ) )
48	44, 47	pm2.61d1 171	. . . . 5 |- ( ( Lim x /\ A. y e. x Tr ( R1 ` y ) ) -> Tr ( R1 ` x ) )
49	48	ex 450	. . . 4 |- ( Lim x -> ( A. y e. x Tr ( R1 ` y ) -> Tr ( R1 ` x ) ) )
50	11, 14, 17, 20, 21, 37, 49	tfinds 7059	. . 3 |- ( A e. On -> Tr ( R1 ` A ) )
51	6, 50	syl 17	. 2 |- ( A e. dom R1 -> Tr ( R1 ` A ) )
52		ndmfv 6218	. . . 4 |- ( -. A e. dom R1 -> ( R1 ` A ) = (/) )
53		treq 4758	. . . 4 |- ( ( R1 ` A ) = (/) -> ( Tr ( R1 ` A ) <-> Tr (/) ) )
54	52, 53	syl 17	. . 3 |- ( -. A e. dom R1 -> ( Tr ( R1 ` A ) <-> Tr (/) ) )
55	21, 54	mpbiri 248	. 2 |- ( -. A e. dom R1 -> Tr ( R1 ` A ) )
56	51, 55	pm2.61i 176	1 |- Tr ( R1 ` A )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U_ciun 4520   Tr wtr 4752   dom cdm 5114   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fun wfun 5882   ` cfv 5888   R1cr1 8625

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-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-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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627

This theorem is referenced by:  r1tr2  8640  r1ordg  8641  r1ord3g  8642  r1ord2  8644  r1sssuc  8646  r1pwss  8647  r1val1  8649  rankwflemb  8656  r1elwf  8659  r1elssi  8668  uniwf  8682  tcrank  8747  ackbij2lem3  9063  r1limwun  9558  tskr1om2  9590