Theorem gchaclem 8951

Description: Lemma for gchac 8954 (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 7421	. . . . . 6 |- Rel ~<_
4	3	brrelex2i 4983	. . . . 5 |- ( A ~<_ C -> C e. _V )
5	2, 4	syl 16	. . . 4 |- ( ph -> C e. _V )
6		canth2g 7570	. . . 4 |- ( C e. _V -> C ~< ~P C )
7		sdomdom 7442	. . . 4 |- ( C ~< ~P C -> C ~<_ ~P C )
8	5, 6, 7	3syl 20	. . 3 |- ( ph -> C ~<_ ~P C )
9		domtr 7467	. . 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 7467	. . . . . . . 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 8470	. . . . . 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 8902	. . . . 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 1219	. . . 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 7568	. . . . 5 |- ( A ~<_ C -> ~P A ~<_ ~P C )
24		domtr 7467	. . . . . 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 1758   _Vcvv 3072   ~Pcpw 3963   class class class wbr 4395  (class class class)co 6195   omcom 6581   ~~ cen 7412   ~<_ cdom 7413   ~< csdm 7414   +c ccda 8442  GCHcgch 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-1o 7025  df-2o 7026  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-wdom 7880  df-card 8215  df-cda 8443  df-gch 8894

This theorem is referenced by: (None)