Theorem gchaclem 9052

Description: Lemma for gchac 9055 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)

Hypotheses

Ref	Expression
gchaclem.1	|- ( ph -> om ~<_ A )
gchaclem.3	|- ( ph -> ~P C e. GCH )
gchaclem.4	|- ( ph -> ( A ~<_ C /\ ( B ~<_ ~P C -> ~P A ~<_ B ) ) )

Assertion

Ref	Expression
gchaclem	|- ( ph -> ( A ~<_ ~P C /\ ( B ~<_ ~P ~P C -> ~P A ~<_ B ) ) )

Proof of Theorem gchaclem

Step	Hyp	Ref	Expression
1		gchaclem.4	. . . 4 |- ( ph -> ( A ~<_ C /\ ( B ~<_ ~P C -> ~P A ~<_ B ) ) )
2	1	simpld 459	. . 3 |- ( ph -> A ~<_ C )
3		reldom 7519	. . . . . 6 |- Rel ~<_
4	3	brrelex2i 5040	. . . . 5 |- ( A ~<_ C -> C e. _V )
5	2, 4	syl 16	. . . 4 |- ( ph -> C e. _V )
6		canth2g 7668	. . . 4 |- ( C e. _V -> C ~< ~P C )
7		sdomdom 7540	. . . 4 |- ( C ~< ~P C -> C ~<_ ~P C )
8	5, 6, 7	3syl 20	. . 3 |- ( ph -> C ~<_ ~P C )
9		domtr 7565	. . 3 |- ( ( A ~<_ C /\ C ~<_ ~P C ) -> A ~<_ ~P C )
10	2, 8, 9	syl2anc 661	. 2 |- ( ph -> A ~<_ ~P C )
11		gchaclem.3	. . . . . 6 |- ( ph -> ~P C e. GCH )
12	11	adantr 465	. . . . 5 |- ( ( ph /\ B ~<_ ~P ~P C ) -> ~P C e. GCH )
13		gchaclem.1	. . . . . . . 8 |- ( ph -> om ~<_ A )
14		domtr 7565	. . . . . . . 8 |- ( ( om ~<_ A /\ A ~<_ C ) -> om ~<_ C )
15	13, 2, 14	syl2anc 661	. . . . . . 7 |- ( ph -> om ~<_ C )
16	15	adantr 465	. . . . . 6 |- ( ( ph /\ B ~<_ ~P ~P C ) -> om ~<_ C )
17		pwcdaidm 8571	. . . . . 6 |- ( om ~<_ C -> ( ~P C +c ~P C ) ~~ ~P C )
18	16, 17	syl 16	. . . . 5 |- ( ( ph /\ B ~<_ ~P ~P C ) -> ( ~P C +c ~P C ) ~~ ~P C )
19		simpr 461	. . . . 5 |- ( ( ph /\ B ~<_ ~P ~P C ) -> B ~<_ ~P ~P C )
20		gchdomtri 9003	. . . . 5 |- ( ( ~P C e. GCH /\ ( ~P C +c ~P C ) ~~ ~P C /\ B ~<_ ~P ~P C ) -> ( ~P C ~<_ B \/ B ~<_ ~P C ) )
21	12, 18, 19, 20	syl3anc 1228	. . . 4 |- ( ( ph /\ B ~<_ ~P ~P C ) -> ( ~P C ~<_ B \/ B ~<_ ~P C ) )
22	21	ex 434	. . 3 |- ( ph -> ( B ~<_ ~P ~P C -> ( ~P C ~<_ B \/ B ~<_ ~P C ) ) )
23		pwdom 7666	. . . . 5 |- ( A ~<_ C -> ~P A ~<_ ~P C )
24		domtr 7565	. . . . . 6 |- ( ( ~P A ~<_ ~P C /\ ~P C ~<_ B ) -> ~P A ~<_ B )
25	24	ex 434	. . . . 5 |- ( ~P A ~<_ ~P C -> ( ~P C ~<_ B -> ~P A ~<_ B ) )
26	2, 23, 25	3syl 20	. . . 4 |- ( ph -> ( ~P C ~<_ B -> ~P A ~<_ B ) )
27	1	simprd 463	. . . 4 |- ( ph -> ( B ~<_ ~P C -> ~P A ~<_ B ) )
28	26, 27	jaod 380	. . 3 |- ( ph -> ( ( ~P C ~<_ B \/ B ~<_ ~P C ) -> ~P A ~<_ B ) )
29	22, 28	syld 44	. 2 |- ( ph -> ( B ~<_ ~P ~P C -> ~P A ~<_ B ) )
30	10, 29	jca 532	1 |- ( ph -> ( A ~<_ ~P C /\ ( B ~<_ ~P ~P C -> ~P A ~<_ B ) ) )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   e. wcel 1767   _Vcvv 3113   ~Pcpw 4010   class class class wbr 4447  (class class class)co 6282   omcom 6678   ~~ cen 7510   ~<_ cdom 7511   ~< csdm 7512   +c ccda 8543  GCHcgch 8994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-2o 7128  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-wdom 7981  df-card 8316  df-cda 8544  df-gch 8995

This theorem is referenced by: (None)