Theorem gchina 9521

Description: Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.)

Assertion

Ref	Expression
gchina	|- (GCH = _V -> InaccW = Inacc )

Proof of Theorem gchina

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

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( (GCH = _V /\ x e. InaccW ) -> x e. InaccW )
2		idd 24	. . . . . . 7 |- ( (GCH = _V /\ x e. InaccW ) -> ( x =/= (/) -> x =/= (/) ) )
3		idd 24	. . . . . . 7 |- ( (GCH = _V /\ x e. InaccW ) -> ( ( cf ` x ) = x -> ( cf ` x ) = x ) )
4		pwfi 8261	. . . . . . . . . . . . 13 |- ( y e. Fin <-> ~P y e. Fin )
5		isfinite 8549	. . . . . . . . . . . . . 14 |- ( ~P y e. Fin <-> ~P y ~< om )
6		winainf 9516	. . . . . . . . . . . . . . . 16 |- ( x e. InaccW -> om C_ x )
7		ssdomg 8001	. . . . . . . . . . . . . . . 16 |- ( x e. InaccW -> ( om C_ x -> om ~<_ x ) )
8	6, 7	mpd 15	. . . . . . . . . . . . . . 15 |- ( x e. InaccW -> om ~<_ x )
9		sdomdomtr 8093	. . . . . . . . . . . . . . . 16 |- ( ( ~P y ~< om /\ om ~<_ x ) -> ~P y ~< x )
10	9	expcom 451	. . . . . . . . . . . . . . 15 |- ( om ~<_ x -> ( ~P y ~< om -> ~P y ~< x ) )
11	8, 10	syl 17	. . . . . . . . . . . . . 14 |- ( x e. InaccW -> ( ~P y ~< om -> ~P y ~< x ) )
12	5, 11	syl5bi 232	. . . . . . . . . . . . 13 |- ( x e. InaccW -> ( ~P y e. Fin -> ~P y ~< x ) )
13	4, 12	syl5bi 232	. . . . . . . . . . . 12 |- ( x e. InaccW -> ( y e. Fin -> ~P y ~< x ) )
14	13	ad3antlr 767	. . . . . . . . . . 11 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ z e. x ) -> ( y e. Fin -> ~P y ~< x ) )
15	14	a1dd 50	. . . . . . . . . 10 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ z e. x ) -> ( y e. Fin -> ( y ~< z -> ~P y ~< x ) ) )
16		vex 3203	. . . . . . . . . . . . . . 15 |- y e. _V
17		simplll 798	. . . . . . . . . . . . . . 15 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ ( z e. x /\ -. y e. Fin ) ) -> GCH = _V )
18	16, 17	syl5eleqr 2708	. . . . . . . . . . . . . 14 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ ( z e. x /\ -. y e. Fin ) ) -> y e. GCH )
19		simprr 796	. . . . . . . . . . . . . 14 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ ( z e. x /\ -. y e. Fin ) ) -> -. y e. Fin )
20		gchinf 9479	. . . . . . . . . . . . . 14 |- ( ( y e. GCH /\ -. y e. Fin ) -> om ~<_ y )
21	18, 19, 20	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ ( z e. x /\ -. y e. Fin ) ) -> om ~<_ y )
22		vex 3203	. . . . . . . . . . . . . 14 |- z e. _V
23	22, 17	syl5eleqr 2708	. . . . . . . . . . . . 13 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ ( z e. x /\ -. y e. Fin ) ) -> z e. GCH )
24		gchpwdom 9492	. . . . . . . . . . . . 13 |- ( ( om ~<_ y /\ y e. GCH /\ z e. GCH ) -> ( y ~< z <-> ~P y ~<_ z ) )
25	21, 18, 23, 24	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ ( z e. x /\ -. y e. Fin ) ) -> ( y ~< z <-> ~P y ~<_ z ) )
26		winacard 9514	. . . . . . . . . . . . . . . . 17 |- ( x e. InaccW -> ( card ` x ) = x )
27		iscard 8801	. . . . . . . . . . . . . . . . . 18 |- ( ( card ` x ) = x <-> ( x e. On /\ A. z e. x z ~< x ) )
28	27	simprbi 480	. . . . . . . . . . . . . . . . 17 |- ( ( card ` x ) = x -> A. z e. x z ~< x )
29	26, 28	syl 17	. . . . . . . . . . . . . . . 16 |- ( x e. InaccW -> A. z e. x z ~< x )
30	29	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) -> A. z e. x z ~< x )
31	30	r19.21bi 2932	. . . . . . . . . . . . . 14 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ z e. x ) -> z ~< x )
32		domsdomtr 8095	. . . . . . . . . . . . . . 15 |- ( ( ~P y ~<_ z /\ z ~< x ) -> ~P y ~< x )
33	32	expcom 451	. . . . . . . . . . . . . 14 |- ( z ~< x -> ( ~P y ~<_ z -> ~P y ~< x ) )
34	31, 33	syl 17	. . . . . . . . . . . . 13 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ z e. x ) -> ( ~P y ~<_ z -> ~P y ~< x ) )
35	34	adantrr 753	. . . . . . . . . . . 12 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ ( z e. x /\ -. y e. Fin ) ) -> ( ~P y ~<_ z -> ~P y ~< x ) )
36	25, 35	sylbid 230	. . . . . . . . . . 11 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ ( z e. x /\ -. y e. Fin ) ) -> ( y ~< z -> ~P y ~< x ) )
37	36	expr 643	. . . . . . . . . 10 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ z e. x ) -> ( -. y e. Fin -> ( y ~< z -> ~P y ~< x ) ) )
38	15, 37	pm2.61d 170	. . . . . . . . 9 |- ( ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) /\ z e. x ) -> ( y ~< z -> ~P y ~< x ) )
39	38	rexlimdva 3031	. . . . . . . 8 |- ( ( (GCH = _V /\ x e. InaccW ) /\ y e. x ) -> ( E. z e. x y ~< z -> ~P y ~< x ) )
40	39	ralimdva 2962	. . . . . . 7 |- ( (GCH = _V /\ x e. InaccW ) -> ( A. y e. x E. z e. x y ~< z -> A. y e. x ~P y ~< x ) )
41	2, 3, 40	3anim123d 1406	. . . . . 6 |- ( (GCH = _V /\ x e. InaccW ) -> ( ( x =/= (/) /\ ( cf ` x ) = x /\ A. y e. x E. z e. x y ~< z ) -> ( x =/= (/) /\ ( cf ` x ) = x /\ A. y e. x ~P y ~< x ) ) )
42		elwina 9508	. . . . . 6 |- ( x e. InaccW <-> ( x =/= (/) /\ ( cf ` x ) = x /\ A. y e. x E. z e. x y ~< z ) )
43		elina 9509	. . . . . 6 |- ( x e. Inacc <-> ( x =/= (/) /\ ( cf ` x ) = x /\ A. y e. x ~P y ~< x ) )
44	41, 42, 43	3imtr4g 285	. . . . 5 |- ( (GCH = _V /\ x e. InaccW ) -> ( x e. InaccW -> x e. Inacc ) )
45	1, 44	mpd 15	. . . 4 |- ( (GCH = _V /\ x e. InaccW ) -> x e. Inacc )
46	45	ex 450	. . 3 |- (GCH = _V -> ( x e. InaccW -> x e. Inacc ) )
47		inawina 9512	. . 3 |- ( x e. Inacc -> x e. InaccW )
48	46, 47	impbid1 215	. 2 |- (GCH = _V -> ( x e. InaccW <-> x e. Inacc ) )
49	48	eqrdv 2620	1 |- (GCH = _V -> InaccW = Inacc )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   Oncon0 5723   ` cfv 5888   omcom 7065   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   cardccrd 8761   cfccf 8763  GCHcgch 9442   InaccWcwina 9504   Inacccina 9505

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-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-har 8463  df-wdom 8464  df-cnf 8559  df-card 8765  df-cf 8767  df-cda 8990  df-fin4 9109  df-gch 9443  df-wina 9506  df-ina 9507

This theorem is referenced by: (None)