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Theorem nosepdmlem 31833
Description: Lemma for nosepdm 31834. (Contributed by Scott Fenton, 24-Nov-2021.)
Assertion
Ref Expression
nosepdmlem |- ( ( A e. No /\ B e. No /\ A <s B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) )
Distinct variable groups:   x, A   x, B
Proof of Theorem nosepdmlem
StepHypRef Expression
1 sltval2 31809 . . . . 5 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) )
2 fvex 6201 . . . . . . 7 |- ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) e. _V
3 fvex 6201 . . . . . . 7 |- ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) e. _V
42, 3brtp 31639 . . . . . 6 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) <-> ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
5 df-3or 1038 . . . . . . . . . 10 |- ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) <-> ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
6 ndmfv 6218 . . . . . . . . . . . . 13 |- ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) )
7 1on 7567 . . . . . . . . . . . . . . . . . . . 20 |- 1o e. On
87elexi 3213 . . . . . . . . . . . . . . . . . . 19 |- 1o e. _V
98prid1 4297 . . . . . . . . . . . . . . . . . 18 |- 1o e. { 1o , 2o }
109nosgnn0i 31812 . . . . . . . . . . . . . . . . 17 |- (/) =/= 1o
11 neeq1 2856 . . . . . . . . . . . . . . . . 17 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) -> ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= 1o <-> (/) =/= 1o ) )
1210, 11mpbiri 248 . . . . . . . . . . . . . . . 16 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) -> ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= 1o )
1312neneqd 2799 . . . . . . . . . . . . . . 15 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) -> -. ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o )
1413intnanrd 963 . . . . . . . . . . . . . 14 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) -> -. ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) )
1513intnanrd 963 . . . . . . . . . . . . . 14 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) -> -. ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) )
16 ioran 511 . . . . . . . . . . . . . 14 |- ( -. ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) <-> ( -. ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) /\ -. ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
1714, 15, 16sylanbrc 698 . . . . . . . . . . . . 13 |- ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) -> -. ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
186, 17syl 17 . . . . . . . . . . . 12 |- ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A -> -. ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
1918adantl 482 . . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No ) /\ -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) -> -. ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
20 orel1 397 . . . . . . . . . . 11 |- ( -. ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> ( ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
2119, 20syl 17 . . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No ) /\ -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) -> ( ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
225, 21syl5bi 232 . . . . . . . . 9 |- ( ( ( A e. No /\ B e. No ) /\ -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) -> ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) )
23 ndmfv 6218 . . . . . . . . . . . . 13 |- ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) )
24 2on 7568 . . . . . . . . . . . . . . . . 17 |- 2o e. On
2524elexi 3213 . . . . . . . . . . . . . . . 16 |- 2o e. _V
2625prid2 4298 . . . . . . . . . . . . . . 15 |- 2o e. { 1o , 2o }
2726nosgnn0i 31812 . . . . . . . . . . . . . 14 |- (/) =/= 2o
28 neeq1 2856 . . . . . . . . . . . . . 14 |- ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) -> ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= 2o <-> (/) =/= 2o ) )
2927, 28mpbiri 248 . . . . . . . . . . . . 13 |- ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= 2o )
3023, 29syl 17 . . . . . . . . . . . 12 |- ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B -> ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= 2o )
3130neneqd 2799 . . . . . . . . . . 11 |- ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B -> -. ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o )
3231con4i 113 . . . . . . . . . 10 |- ( ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B )
3332adantl 482 . . . . . . . . 9 |- ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B )
3422, 33syl6 35 . . . . . . . 8 |- ( ( ( A e. No /\ B e. No ) /\ -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A ) -> ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) )
3534ex 450 . . . . . . 7 |- ( ( A e. No /\ B e. No ) -> ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A -> ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) ) )
3635com23 86 . . . . . 6 |- ( ( A e. No /\ B e. No ) -> ( ( ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) \/ ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = (/) /\ ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 2o ) ) -> ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) ) )
374, 36syl5bi 232 . . . . 5 |- ( ( A e. No /\ B e. No ) -> ( ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) ) )
381, 37sylbid 230 . . . 4 |- ( ( A e. No /\ B e. No ) -> ( A <s B -> ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) ) )
39383impia 1261 . . 3 |- ( ( A e. No /\ B e. No /\ A <s B ) -> ( -. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) )
4039orrd 393 . 2 |- ( ( A e. No /\ B e. No /\ A <s B ) -> ( |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A \/ |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) )
41 elun 3753 . 2 |- ( |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) <-> ( |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom A \/ |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. dom B ) )
4240, 41sylibr 224 1 |- ( ( A e. No /\ B e. No /\ A <s B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   u. cun 3572   (/)c0 3915   {ctp 4181   <.cop 4183   |^|cint 4475   class class class wbr 4653   dom cdm 5114   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-dm 5124  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:  nosepdm  31834
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