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Theorem nogesgn1o 27719
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 1192 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o)) → 𝐵 No )
2 nofv 27703 . . . . . 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 1485 . . . . 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 1205 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝑋 ∈ On)
7 fveq1 6904 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87adantr 480 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
9 simpr 484 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
109fvresd 6925 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
119fvresd 6925 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
128, 10, 113eqtr3d 2784 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐴𝑦) = (𝐵𝑦))
1312ralrimiva 3145 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
1413adantr 480 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
15143ad2ant2 1134 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
16 simp2r 1200 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴𝑋) = 1o)
17 simp3 1138 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o))
1816, 17jca 511 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → ((𝐴𝑋) = 1o ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)))
19 andi 1009 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) ↔ (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o)))
2018, 19sylib 218 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o)))
21 3mix1 1330 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
22 3mix2 1331 . . . . . . . . . 10 (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) → (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2321, 22jaoi 857 . . . . . . . . 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 6918 . . . . . . . . 9 (𝐴𝑋) ∈ V
26 fvex 6918 . . . . . . . . 9 (𝐵𝑋) ∈ V
2725, 26brtp 5527 . . . . . . . 8 ((𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋) ↔ (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2824, 27sylibr 234 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))
29 raleq 3322 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦)))
30 fveq2 6905 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
31 fveq2 6905 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
3230, 31breq12d 5155 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
3329, 32anbi12d 632 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ (∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))))
3433rspcev 3621 . . . . . . 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 836 . . . . . 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 1203 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐴 No )
37 simp12 1204 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐵 No )
38 sltval 27693 . . . . . . 7 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
3936, 37, 38syl2anc 584 . . . . . 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 257 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → 𝐴 <s 𝐵)
41403expia 1121 . . . 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 1117 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 206  wa 395  wo 847  w3o 1085  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  c0 4332  {ctp 4629  cop 4631   class class class wbr 5142  cres 5686  Oncon0 6383  cfv 6560  1oc1o 8500  2oc2o 8501   No csur 27685   <s cslt 27686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689
This theorem is referenced by:  nogesgn1ores  27720
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