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Theorem nolesgn2o 33286
 Description: Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2o, then 𝐵(𝑋) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
nolesgn2o (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2o)

Proof of Theorem nolesgn2o
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) = 2o)) → 𝐵 No )
2 nofv 33272 . . . . . 6 (𝐵 No → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o))
31, 2syl 17 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o))
4 3orel3 33050 . . . . 5 (¬ (𝐵𝑋) = 2o → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)))
53, 4syl5com 31 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (¬ (𝐵𝑋) = 2o → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)))
6 simp13 1202 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝑋 ∈ On)
7 fveq1 6648 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87eqcomd 2807 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
98adantr 484 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
10 simpr 488 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
1110fvresd 6669 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
1210fvresd 6669 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
139, 11, 123eqtr3d 2844 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐵𝑦) = (𝐴𝑦))
1413ralrimiva 3152 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
1514adantr 484 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
16153ad2ant2 1131 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
17 simprr 772 . . . . . . . . . . . . 13 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (𝐴𝑋) = 2o)
1817a1d 25 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = ∅ → (𝐴𝑋) = 2o))
1918ancld 554 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = ∅ → ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
2017a1d 25 . . . . . . . . . . . 12 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = 1o → (𝐴𝑋) = 2o))
2120ancld 554 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → ((𝐵𝑋) = 1o → ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o)))
2219, 21orim12d 962 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o))))
23223impia 1114 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o)))
24 3mix3 1329 . . . . . . . . . 10 (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
25 3mix2 1328 . . . . . . . . . 10 (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
2624, 25jaoi 854 . . . . . . . . 9 ((((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o)) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
2723, 26syl 17 . . . . . . . 8 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
28 fvex 6662 . . . . . . . . 9 (𝐵𝑋) ∈ V
29 fvex 6662 . . . . . . . . 9 (𝐴𝑋) ∈ V
3028, 29brtp 33093 . . . . . . . 8 ((𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋) ↔ (((𝐵𝑋) = 1o ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1o ∧ (𝐴𝑋) = 2o) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2o)))
3127, 30sylibr 237 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋))
32 raleq 3361 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ↔ ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦)))
33 fveq2 6649 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
34 fveq2 6649 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
3533, 34breq12d 5046 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥) ↔ (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋)))
3632, 35anbi12d 633 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥)) ↔ (∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋))))
3736rspcev 3574 . . . . . . 7 ((𝑋 ∈ On ∧ (∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑋))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥)))
386, 16, 31, 37syl12anc 835 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥)))
39 simp12 1201 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝐵 No )
40 simp11 1200 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝐴 No )
41 sltval 33262 . . . . . . 7 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥))))
4239, 40, 41syl2anc 587 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴𝑥))))
4338, 42mpbird 260 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o)) → 𝐵 <s 𝐴)
44433expia 1118 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o) → 𝐵 <s 𝐴))
455, 44syld 47 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (¬ (𝐵𝑋) = 2o𝐵 <s 𝐴))
4645con1d 147 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o)) → (¬ 𝐵 <s 𝐴 → (𝐵𝑋) = 2o))
47463impia 1114 1 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2o) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2o)
 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 2112  ∀wral 3109  ∃wrex 3110  ∅c0 4246  {ctp 4532  ⟨cop 4534   class class class wbr 5033   ↾ cres 5525  Oncon0 6163  ‘cfv 6328  1oc1o 8082  2oc2o 8083   No csur 33255
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