Theorem gchaclem 9500

Description: Lemma for gchac 9503 (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 475	. . 3 |- ( ph -> A ~<_ C )
3		reldom 7961	. . . . . 6 |- Rel ~<_
4	3	brrelex2i 5159	. . . . 5 |- ( A ~<_ C -> C e. _V )
5	2, 4	syl 17	. . . 4 |- ( ph -> C e. _V )
6		canth2g 8114	. . . 4 |- ( C e. _V -> C ~< ~P C )
7		sdomdom 7983	. . . 4 |- ( C ~< ~P C -> C ~<_ ~P C )
8	5, 6, 7	3syl 18	. . 3 |- ( ph -> C ~<_ ~P C )
9		domtr 8009	. . 3 |- ( ( A ~<_ C /\ C ~<_ ~P C ) -> A ~<_ ~P C )
10	2, 8, 9	syl2anc 693	. 2 |- ( ph -> A ~<_ ~P C )
11		gchaclem.3	. . . . . 6 |- ( ph -> ~P C e. GCH )
12	11	adantr 481	. . . . 5 |- ( ( ph /\ B ~<_ ~P ~P C ) -> ~P C e. GCH )
13		gchaclem.1	. . . . . . . 8 |- ( ph -> om ~<_ A )
14		domtr 8009	. . . . . . . 8 |- ( ( om ~<_ A /\ A ~<_ C ) -> om ~<_ C )
15	13, 2, 14	syl2anc 693	. . . . . . 7 |- ( ph -> om ~<_ C )
16	15	adantr 481	. . . . . 6 |- ( ( ph /\ B ~<_ ~P ~P C ) -> om ~<_ C )
17		pwcdaidm 9017	. . . . . 6 |- ( om ~<_ C -> ( ~P C +c ~P C ) ~~ ~P C )
18	16, 17	syl 17	. . . . 5 |- ( ( ph /\ B ~<_ ~P ~P C ) -> ( ~P C +c ~P C ) ~~ ~P C )
19		simpr 477	. . . . 5 |- ( ( ph /\ B ~<_ ~P ~P C ) -> B ~<_ ~P ~P C )
20		gchdomtri 9451	. . . . 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 1326	. . . 4 |- ( ( ph /\ B ~<_ ~P ~P C ) -> ( ~P C ~<_ B \/ B ~<_ ~P C ) )
22	21	ex 450	. . 3 |- ( ph -> ( B ~<_ ~P ~P C -> ( ~P C ~<_ B \/ B ~<_ ~P C ) ) )
23		pwdom 8112	. . . . 5 |- ( A ~<_ C -> ~P A ~<_ ~P C )
24		domtr 8009	. . . . . 6 |- ( ( ~P A ~<_ ~P C /\ ~P C ~<_ B ) -> ~P A ~<_ B )
25	24	ex 450	. . . . 5 |- ( ~P A ~<_ ~P C -> ( ~P C ~<_ B -> ~P A ~<_ B ) )
26	2, 23, 25	3syl 18	. . . 4 |- ( ph -> ( ~P C ~<_ B -> ~P A ~<_ B ) )
27	1	simprd 479	. . . 4 |- ( ph -> ( B ~<_ ~P C -> ~P A ~<_ B ) )
28	26, 27	jaod 395	. . 3 |- ( ph -> ( ( ~P C ~<_ B \/ B ~<_ ~P C ) -> ~P A ~<_ B ) )
29	22, 28	syld 47	. 2 |- ( ph -> ( B ~<_ ~P ~P C -> ~P A ~<_ B ) )
30	10, 29	jca 554	1 |- ( ph -> ( A ~<_ ~P C /\ ( B ~<_ ~P ~P C -> ~P A ~<_ B ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653  (class class class)co 6650   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   +c ccda 8989  GCHcgch 9442

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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wdom 8464  df-card 8765  df-cda 8990  df-gch 9443

This theorem is referenced by: (None)