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Theorem nogesgn1o 27624
Description: Given 𝐴 greater than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 1o, then 𝐵(𝑋) = 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
Assertion
Ref Expression
nogesgn1o (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵𝑋) = 1o)

Proof of Theorem nogesgn1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1189 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → 𝐵 No )
2 nofv 27608 . . . . . 6 (𝐵 No → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o))
31, 2syl 17 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o))
4 3orel2 1480 . . . . 5 (¬ (𝐵𝑋) = 1o → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)))
53, 4syl5com 31 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → (¬ (𝐵𝑋) = 1o → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)))
6 simp13 1202 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝑋 ∈ On)
7 fveq1 6891 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87adantr 479 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
9 simpr 483 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
109fvresd 6912 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
119fvresd 6912 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
128, 10, 113eqtr3d 2773 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐴𝑦) = (𝐵𝑦))
1312ralrimiva 3136 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
1413adantr 479 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
15143ad2ant2 1131 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
16 simp2r 1197 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴𝑋) = 1o)
17 simp3 1135 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o))
1816, 17jca 510 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ((𝐴𝑋) = 1o ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)))
19 andi 1005 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) ↔ (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o)))
2018, 19sylib 217 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o)))
21 3mix1 1327 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
22 3mix2 1328 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2321, 22jaoi 855 . . . . . . . . 9 ((((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o)) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2420, 23syl 17 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
25 fvex 6905 . . . . . . . . 9 (𝐴𝑋) ∈ V
26 fvex 6905 . . . . . . . . 9 (𝐵𝑋) ∈ V
2725, 26brtp 5519 . . . . . . . 8 ((𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋) ↔ (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2824, 27sylibr 233 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))
29 raleq 3312 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦)))
30 fveq2 6892 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
31 fveq2 6892 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
3230, 31breq12d 5156 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
3329, 32anbi12d 630 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ (∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))))
3433rspcev 3601 . . . . . . 7 ((𝑋 ∈ On ∧ (∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
356, 15, 28, 34syl12anc 835 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
36 simp11 1200 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐴 No )
37 simp12 1201 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐵 No )
38 sltval 27598 . . . . . . 7 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
3936, 37, 38syl2anc 582 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
4035, 39mpbird 256 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐴 <s 𝐵)
41403expia 1118 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o) → 𝐴 <s 𝐵))
425, 41syld 47 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → (¬ (𝐵𝑋) = 1o𝐴 <s 𝐵))
4342con1d 145 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → (¬ 𝐴 <s 𝐵 → (𝐵𝑋) = 1o))
44433impia 1114 1 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵𝑋) = 1o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3o 1083  w3a 1084   = wceq 1533  wcel 2098  wral 3051  wrex 3060  c0 4318  {ctp 4628  cop 4630   class class class wbr 5143  cres 5674  Oncon0 6364  cfv 6543  1oc1o 8478  2oc2o 8479   No csur 27591   <s cslt 27592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1o 8485  df-2o 8486  df-no 27594  df-slt 27595
This theorem is referenced by:  nogesgn1ores  27625
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