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Theorem nogesgn1o 33441
 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 33425 . . . . . 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 33172 . . . . 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 6657 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87adantr 484 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
9 simpr 488 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
109fvresd 6678 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
119fvresd 6678 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
128, 10, 113eqtr3d 2801 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐴𝑦) = (𝐵𝑦))
1312ralrimiva 3113 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦))
1413adantr 484 . . . . . . . 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 515 . . . . . . . . . 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 221 . . . . . . . . 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 854 . . . . . . . . 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 6671 . . . . . . . . 9 (𝐴𝑋) ∈ V
26 fvex 6671 . . . . . . . . 9 (𝐵𝑋) ∈ V
2725, 26brtp 33232 . . . . . . . 8 ((𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋) ↔ (((𝐴𝑋) = 1o ∧ (𝐵𝑋) = ∅) ∨ ((𝐴𝑋) = 1o ∧ (𝐵𝑋) = 2o) ∨ ((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 2o)))
2824, 27sylibr 237 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 1o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 2o)) → (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))
29 raleq 3323 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ↔ ∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦)))
30 fveq2 6658 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
31 fveq2 6658 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
3230, 31breq12d 5045 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
3329, 32anbi12d 633 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ (∀𝑦𝑋 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))))
3433rspcev 3541 . . . . . . 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 33415 . . . . . . 7 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
3936, 37, 38syl2anc 587 . . . . . 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 260 . . . . 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 147 . 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 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ∅c0 4225  {ctp 4526  ⟨cop 4528   class class class wbr 5032   ↾ cres 5526  Oncon0 6169  ‘cfv 6335  1oc1o 8105  2oc2o 8106   No csur 33408
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