Theorem inawinalem 9511

Description: Lemma for inawina 9512. (Contributed by Mario Carneiro, 8-Jun-2014.)

Assertion

Ref	Expression
inawinalem	|- ( A e. On -> ( A. x e. A ~P x ~< A -> A. x e. A E. y e. A x ~< y ) )

Distinct variable group:   x, A, y

Proof of Theorem inawinalem

Step	Hyp	Ref	Expression
1		sdomdom 7983	. . . . 5 |- ( ~P x ~< A -> ~P x ~<_ A )
2		ondomen 8860	. . . . . 6 |- ( ( A e. On /\ ~P x ~<_ A ) -> ~P x e. dom card )
3		isnum2 8771	. . . . . 6 |- ( ~P x e. dom card <-> E. y e. On y ~~ ~P x )
4	2, 3	sylib 208	. . . . 5 |- ( ( A e. On /\ ~P x ~<_ A ) -> E. y e. On y ~~ ~P x )
5	1, 4	sylan2 491	. . . 4 |- ( ( A e. On /\ ~P x ~< A ) -> E. y e. On y ~~ ~P x )
6		ensdomtr 8096	. . . . . . . . 9 |- ( ( y ~~ ~P x /\ ~P x ~< A ) -> y ~< A )
7	6	ad2ant2l 782	. . . . . . . 8 |- ( ( ( y e. On /\ y ~~ ~P x ) /\ ( A e. On /\ ~P x ~< A ) ) -> y ~< A )
8		sdomel 8107	. . . . . . . . 9 |- ( ( y e. On /\ A e. On ) -> ( y ~< A -> y e. A ) )
9	8	ad2ant2r 783	. . . . . . . 8 |- ( ( ( y e. On /\ y ~~ ~P x ) /\ ( A e. On /\ ~P x ~< A ) ) -> ( y ~< A -> y e. A ) )
10	7, 9	mpd 15	. . . . . . 7 |- ( ( ( y e. On /\ y ~~ ~P x ) /\ ( A e. On /\ ~P x ~< A ) ) -> y e. A )
11		vex 3203	. . . . . . . . . 10 |- x e. _V
12	11	canth2 8113	. . . . . . . . 9 |- x ~< ~P x
13		ensym 8005	. . . . . . . . 9 |- ( y ~~ ~P x -> ~P x ~~ y )
14		sdomentr 8094	. . . . . . . . 9 |- ( ( x ~< ~P x /\ ~P x ~~ y ) -> x ~< y )
15	12, 13, 14	sylancr 695	. . . . . . . 8 |- ( y ~~ ~P x -> x ~< y )
16	15	ad2antlr 763	. . . . . . 7 |- ( ( ( y e. On /\ y ~~ ~P x ) /\ ( A e. On /\ ~P x ~< A ) ) -> x ~< y )
17	10, 16	jca 554	. . . . . 6 |- ( ( ( y e. On /\ y ~~ ~P x ) /\ ( A e. On /\ ~P x ~< A ) ) -> ( y e. A /\ x ~< y ) )
18	17	expcom 451	. . . . 5 |- ( ( A e. On /\ ~P x ~< A ) -> ( ( y e. On /\ y ~~ ~P x ) -> ( y e. A /\ x ~< y ) ) )
19	18	reximdv2 3014	. . . 4 |- ( ( A e. On /\ ~P x ~< A ) -> ( E. y e. On y ~~ ~P x -> E. y e. A x ~< y ) )
20	5, 19	mpd 15	. . 3 |- ( ( A e. On /\ ~P x ~< A ) -> E. y e. A x ~< y )
21	20	ex 450	. 2 |- ( A e. On -> ( ~P x ~< A -> E. y e. A x ~< y ) )
22	21	ralimdv 2963	1 |- ( A e. On -> ( A. x e. A ~P x ~< A -> A. x e. A E. y e. A x ~< y ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761

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

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-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-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-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765

This theorem is referenced by:  inawina  9512  tskcard  9603  gruina  9640