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Theorem nolesgn2o 32152
 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 1230 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → 𝐵 No )
2 nofv 32138 . . . . . 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 31922 . . . . 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 1248 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝑋 ∈ On)
7 fveq1 6353 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋)‘𝑦) = ((𝐵𝑋)‘𝑦))
87eqcomd 2767 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
98adantr 472 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = ((𝐴𝑋)‘𝑦))
10 simpr 479 . . . . . . . . . . . 12 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
1110fvresd 6371 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐵𝑋)‘𝑦) = (𝐵𝑦))
1210fvresd 6371 . . . . . . . . . . 11 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → ((𝐴𝑋)‘𝑦) = (𝐴𝑦))
139, 11, 123eqtr3d 2803 . . . . . . . . . 10 (((𝐴𝑋) = (𝐵𝑋) ∧ 𝑦𝑋) → (𝐵𝑦) = (𝐴𝑦))
1413ralrimiva 3105 . . . . . . . . 9 ((𝐴𝑋) = (𝐵𝑋) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
1514adantr 472 . . . . . . . 8 (((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
16153ad2ant2 1129 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦))
17 simprr 813 . . . . . . . . . . . . 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 577 . . . . . . . . . . 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 577 . . . . . . . . . . 11 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → ((𝐵𝑋) = 1𝑜 → ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜)))
2219, 21orim12d 919 . . . . . . . . . 10 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜))))
23223impia 1110 . . . . . . . . 9 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜)))
24 3mix3 1417 . . . . . . . . . 10 (((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜) → (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜)))
25 3mix2 1416 . . . . . . . . . 10 (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) → (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜)))
2624, 25jaoi 393 . . . . . . . . 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 6364 . . . . . . . . 9 (𝐵𝑋) ∈ V
29 fvex 6364 . . . . . . . . 9 (𝐴𝑋) ∈ V
3028, 29brtp 31968 . . . . . . . 8 ((𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋) ↔ (((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = ∅) ∨ ((𝐵𝑋) = 1𝑜 ∧ (𝐴𝑋) = 2𝑜) ∨ ((𝐵𝑋) = ∅ ∧ (𝐴𝑋) = 2𝑜)))
3127, 30sylibr 224 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → (𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋))
32 raleq 3278 . . . . . . . . 9 (𝑥 = 𝑋 → (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ↔ ∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦)))
33 fveq2 6354 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
34 fveq2 6354 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
3533, 34breq12d 4818 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥) ↔ (𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋)))
3632, 35anbi12d 749 . . . . . . . 8 (𝑥 = 𝑋 → ((∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥)) ↔ (∀𝑦𝑋 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑋))))
3736rspcev 3450 . . . . . . 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 1475 . . . . . 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 1247 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝐵 No )
40 simp11 1246 . . . . . . 7 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝐴 No )
41 sltval 32128 . . . . . . 7 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐵𝑦) = (𝐴𝑦) ∧ (𝐵𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴𝑥))))
4239, 40, 41syl2anc 696 . . . . . 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 247 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜)) → 𝐵 <s 𝐴)
44433expia 1115 . . . 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 139 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜)) → (¬ 𝐵 <s 𝐴 → (𝐵𝑋) = 2𝑜))
47463impia 1110 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 196   ∨ wo 382   ∧ wa 383   ∨ w3o 1071   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  ∀wral 3051  ∃wrex 3052  ∅c0 4059  {ctp 4326  ⟨cop 4328   class class class wbr 4805   ↾ cres 5269  Oncon0 5885  ‘cfv 6050  1𝑜c1o 7724  2𝑜c2o 7725   No csur 32121
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