Theorem gchaclem 9045

Description: Lemma for gchac 9048 (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 457	. . 3 |- ( ph -> A ~<_ C )
3		reldom 7515	. . . . . 6 |- Rel ~<_
4	3	brrelex2i 5030	. . . . 5 |- ( A ~<_ C -> C e. _V )
5	2, 4	syl 16	. . . 4 |- ( ph -> C e. _V )
6		canth2g 7664	. . . 4 |- ( C e. _V -> C ~< ~P C )
7		sdomdom 7536	. . . 4 |- ( C ~< ~P C -> C ~<_ ~P C )
8	5, 6, 7	3syl 20	. . 3 |- ( ph -> C ~<_ ~P C )
9		domtr 7561	. . 3 |- ( ( A ~<_ C /\ C ~<_ ~P C ) -> A ~<_ ~P C )
10	2, 8, 9	syl2anc 659	. 2 |- ( ph -> A ~<_ ~P C )
11		gchaclem.3	. . . . . 6 |- ( ph -> ~P C e. GCH )
12	11	adantr 463	. . . . 5 |- ( ( ph /\ B ~<_ ~P ~P C ) -> ~P C e. GCH )
13		gchaclem.1	. . . . . . . 8 |- ( ph -> om ~<_ A )
14		domtr 7561	. . . . . . . 8 |- ( ( om ~<_ A /\ A ~<_ C ) -> om ~<_ C )
15	13, 2, 14	syl2anc 659	. . . . . . 7 |- ( ph -> om ~<_ C )
16	15	adantr 463	. . . . . 6 |- ( ( ph /\ B ~<_ ~P ~P C ) -> om ~<_ C )
17		pwcdaidm 8566	. . . . . 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 459	. . . . 5 |- ( ( ph /\ B ~<_ ~P ~P C ) -> B ~<_ ~P ~P C )
20		gchdomtri 8996	. . . . 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 1226	. . . 4 |- ( ( ph /\ B ~<_ ~P ~P C ) -> ( ~P C ~<_ B \/ B ~<_ ~P C ) )
22	21	ex 432	. . 3 |- ( ph -> ( B ~<_ ~P ~P C -> ( ~P C ~<_ B \/ B ~<_ ~P C ) ) )
23		pwdom 7662	. . . . 5 |- ( A ~<_ C -> ~P A ~<_ ~P C )
24		domtr 7561	. . . . . 6 |- ( ( ~P A ~<_ ~P C /\ ~P C ~<_ B ) -> ~P A ~<_ B )
25	24	ex 432	. . . . 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 461	. . . 4 |- ( ph -> ( B ~<_ ~P C -> ~P A ~<_ B ) )
28	26, 27	jaod 378	. . 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 530	1 |- ( ph -> ( A ~<_ ~P C /\ ( B ~<_ ~P ~P C -> ~P A ~<_ B ) ) )

Syntax hints:   -> wi 4   \/ wo 366   /\ wa 367   e. wcel 1823   _Vcvv 3106   ~Pcpw 3999   class class class wbr 4439  (class class class)co 6270   omcom 6673   ~~ cen 7506   ~<_ cdom 7507   ~< csdm 7508   +c ccda 8538  GCHcgch 8987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-1o 7122  df-2o 7123  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-wdom 7977  df-card 8311  df-cda 8539  df-gch 8988

This theorem is referenced by: (None)