Theorem sltintdifex 31814

Description: If A <s B, then the intersection of all the ordinals that have differing signs in A and B exists. (Contributed by Scott Fenton, 22-Feb-2012.)

Assertion

Ref	Expression
sltintdifex	|- ( ( A e. No /\ B e. No ) -> ( A <s B -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V ) )

Distinct variable groups:   A, a   B, a

Proof of Theorem sltintdifex

Step	Hyp	Ref	Expression
1		sltval2 31809	. 2 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) ) )
2		fvex 6201	. . . 4 |- ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. _V
3		fvex 6201	. . . 4 |- ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) e. _V
4	2, 3	brtp 31639	. . 3 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) <-> ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) )
5		fvprc 6185	. . . . . . 7 |- ( -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V -> ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
6		1n0 7575	. . . . . . . . 9 |- 1o =/= (/)
7	6	neii 2796	. . . . . . . 8 |- -. 1o = (/)
8		eqeq1 2626	. . . . . . . . 9 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o <-> (/) = 1o ) )
9		eqcom 2629	. . . . . . . . 9 |- ( (/) = 1o <-> 1o = (/) )
10	8, 9	syl6bb 276	. . . . . . . 8 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o <-> 1o = (/) ) )
11	7, 10	mtbiri 317	. . . . . . 7 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> -. ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o )
12	5, 11	syl 17	. . . . . 6 |- ( -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V -> -. ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o )
13	12	con4i 113	. . . . 5 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
14	13	adantr 481	. . . 4 |- ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
15	13	adantr 481	. . . 4 |- ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
16		fvprc 6185	. . . . . . 7 |- ( -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V -> ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) )
17		2on0 7569	. . . . . . . . 9 |- 2o =/= (/)
18	17	neii 2796	. . . . . . . 8 |- -. 2o = (/)
19		eqeq1 2626	. . . . . . . . 9 |- ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o <-> (/) = 2o ) )
20		eqcom 2629	. . . . . . . . 9 |- ( (/) = 2o <-> 2o = (/) )
21	19, 20	syl6bb 276	. . . . . . . 8 |- ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o <-> 2o = (/) ) )
22	18, 21	mtbiri 317	. . . . . . 7 |- ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o )
23	16, 22	syl 17	. . . . . 6 |- ( -. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V -> -. ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o )
24	23	con4i 113	. . . . 5 |- ( ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
25	24	adantl 482	. . . 4 |- ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
26	14, 15, 25	3jaoi 1391	. . 3 |- ( ( ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = 2o ) ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
27	4, 26	sylbi 207	. 2 |- ( ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V )
28	1, 27	syl6bi 243	1 |- ( ( A e. No /\ B e. No ) -> ( A <s B -> |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   (/)c0 3915   {ctp 4181   <.cop 4183   |^|cint 4475   class class class wbr 4653   Oncon0 5723   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793   <scslt 31794

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-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fv 5896  df-1o 7560  df-2o 7561  df-slt 31797

This theorem is referenced by:  sltres  31815