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Theorem nolt02o 32072
 Description: Given 𝐴 less than 𝐵, equal to 𝐵 up to 𝑋, and undefined at 𝑋, then 𝐵(𝑋) = 2𝑜. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
nolt02o (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → (𝐵𝑋) = 2𝑜)

Proof of Theorem nolt02o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1222 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐴 No )
2 sltso 32054 . . . . . 6 <s Or No
3 sonr 5160 . . . . . 6 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
42, 3mpan 708 . . . . 5 (𝐴 No → ¬ 𝐴 <s 𝐴)
51, 4syl 17 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ 𝐴 <s 𝐴)
6 simp2r 1219 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐴 <s 𝐵)
7 breq2 4764 . . . . 5 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
86, 7syl5ibrcom 237 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → (𝐴 = 𝐵𝐴 <s 𝐴))
95, 8mtod 189 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ 𝐴 = 𝐵)
10 simpl2l 1259 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = (𝐵𝑋))
11 simpl11 1285 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐴 No )
12 nofun 32029 . . . . . 6 (𝐴 No → Fun 𝐴)
13 funrel 6018 . . . . . 6 (Fun 𝐴 → Rel 𝐴)
1411, 12, 133syl 18 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → Rel 𝐴)
15 simpl13 1287 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝑋 ∈ On)
16 simpl3 1208 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = ∅)
17 nolt02olem 32071 . . . . . 6 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
1811, 15, 16, 17syl3anc 1439 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → dom 𝐴𝑋)
19 relssres 5547 . . . . 5 ((Rel 𝐴 ∧ dom 𝐴𝑋) → (𝐴𝑋) = 𝐴)
2014, 18, 19syl2anc 696 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = 𝐴)
21 simpl12 1286 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐵 No )
22 nofun 32029 . . . . . 6 (𝐵 No → Fun 𝐵)
23 funrel 6018 . . . . . 6 (Fun 𝐵 → Rel 𝐵)
2421, 22, 233syl 18 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → Rel 𝐵)
25 simpr 479 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐵𝑋) = ∅)
26 nolt02olem 32071 . . . . . 6 ((𝐵 No 𝑋 ∈ On ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
2721, 15, 25, 26syl3anc 1439 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
28 relssres 5547 . . . . 5 ((Rel 𝐵 ∧ dom 𝐵𝑋) → (𝐵𝑋) = 𝐵)
2924, 27, 28syl2anc 696 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐵𝑋) = 𝐵)
3010, 20, 293eqtr3d 2766 . . 3 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐴 = 𝐵)
319, 30mtand 694 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ (𝐵𝑋) = ∅)
32 simp12 1223 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐵 No )
33 sltval 32027 . . . . . 6 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
341, 32, 33syl2anc 696 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
356, 34mpbid 222 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
36 df-an 385 . . . . . 6 ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)) ↔ ¬ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
3736rexbii 3143 . . . . 5 (∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)) ↔ ∃𝑥 ∈ On ¬ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
38 rexnal 3097 . . . . 5 (∃𝑥 ∈ On ¬ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)) ↔ ¬ ∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
3937, 38bitri 264 . . . 4 (∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)) ↔ ¬ ∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
4035, 39sylib 208 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ ∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
41 1on 7687 . . . . . . . . . . . . 13 1𝑜 ∈ On
4241elexi 3317 . . . . . . . . . . . 12 1𝑜 ∈ V
4342prid1 4404 . . . . . . . . . . 11 1𝑜 ∈ {1𝑜, 2𝑜}
4443nosgnn0i 32039 . . . . . . . . . 10 ∅ ≠ 1𝑜
4544neii 2898 . . . . . . . . 9 ¬ ∅ = 1𝑜
46 simpll3 1235 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = ∅)
47 simplr 809 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐵𝑋) = 1𝑜)
48 eqeq1 2728 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋) = ∅ ↔ (𝐵𝑋) = ∅))
4948anbi1d 743 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1𝑜) ↔ ((𝐵𝑋) = ∅ ∧ (𝐵𝑋) = 1𝑜)))
50 eqtr2 2744 . . . . . . . . . . . 12 (((𝐵𝑋) = ∅ ∧ (𝐵𝑋) = 1𝑜) → ∅ = 1𝑜)
5149, 50syl6bi 243 . . . . . . . . . . 11 ((𝐴𝑋) = (𝐵𝑋) → (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1𝑜) → ∅ = 1𝑜))
5251com12 32 . . . . . . . . . 10 (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1𝑜) → ((𝐴𝑋) = (𝐵𝑋) → ∅ = 1𝑜))
5346, 47, 52syl2anc 696 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ((𝐴𝑋) = (𝐵𝑋) → ∅ = 1𝑜))
5445, 53mtoi 190 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ (𝐴𝑋) = (𝐵𝑋))
55 simpr 479 . . . . . . . . 9 ((((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) ∧ 𝑋𝑥) → 𝑋𝑥)
56 simplrr 820 . . . . . . . . 9 ((((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) ∧ 𝑋𝑥) → ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))
57 fveq2 6304 . . . . . . . . . . 11 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
58 fveq2 6304 . . . . . . . . . . 11 (𝑦 = 𝑋 → (𝐵𝑦) = (𝐵𝑋))
5957, 58eqeq12d 2739 . . . . . . . . . 10 (𝑦 = 𝑋 → ((𝐴𝑦) = (𝐵𝑦) ↔ (𝐴𝑋) = (𝐵𝑋)))
6059rspcv 3409 . . . . . . . . 9 (𝑋𝑥 → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → (𝐴𝑋) = (𝐵𝑋)))
6155, 56, 60sylc 65 . . . . . . . 8 ((((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) ∧ 𝑋𝑥) → (𝐴𝑋) = (𝐵𝑋))
6254, 61mtand 694 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ 𝑋𝑥)
63 simprl 811 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑥 ∈ On)
64 simpl13 1287 . . . . . . . . 9 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) → 𝑋 ∈ On)
6564adantr 472 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑋 ∈ On)
66 ontri1 5870 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
6763, 65, 66syl2anc 696 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
6862, 67mpbird 247 . . . . . 6 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑥𝑋)
69 onsseleq 5878 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
7063, 65, 69syl2anc 696 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
71 eqtr2 2744 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 1𝑜) → ∅ = 1𝑜)
7271ancoms 468 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = ∅) → ∅ = 1𝑜)
7345, 72mto 188 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = ∅)
74 df-1o 7680 . . . . . . . . . . . . . . . 16 1𝑜 = suc ∅
75 df-2o 7681 . . . . . . . . . . . . . . . 16 2𝑜 = suc 1𝑜
7674, 75eqeq12i 2738 . . . . . . . . . . . . . . 15 (1𝑜 = 2𝑜 ↔ suc ∅ = suc 1𝑜)
77 0elon 5891 . . . . . . . . . . . . . . . 16 ∅ ∈ On
78 suc11 5944 . . . . . . . . . . . . . . . 16 ((∅ ∈ On ∧ 1𝑜 ∈ On) → (suc ∅ = suc 1𝑜 ↔ ∅ = 1𝑜))
7977, 41, 78mp2an 710 . . . . . . . . . . . . . . 15 (suc ∅ = suc 1𝑜 ↔ ∅ = 1𝑜)
8076, 79bitri 264 . . . . . . . . . . . . . 14 (1𝑜 = 2𝑜 ↔ ∅ = 1𝑜)
8144, 80nemtbir 2991 . . . . . . . . . . . . 13 ¬ 1𝑜 = 2𝑜
82 eqtr2 2744 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = 2𝑜) → 1𝑜 = 2𝑜)
8381, 82mto 188 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = 2𝑜)
84 2on 7688 . . . . . . . . . . . . . . . . 17 2𝑜 ∈ On
8584elexi 3317 . . . . . . . . . . . . . . . 16 2𝑜 ∈ V
8685prid2 4405 . . . . . . . . . . . . . . 15 2𝑜 ∈ {1𝑜, 2𝑜}
8786nosgnn0i 32039 . . . . . . . . . . . . . 14 ∅ ≠ 2𝑜
8887neii 2898 . . . . . . . . . . . . 13 ¬ ∅ = 2𝑜
89 eqtr2 2744 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2𝑜) → ∅ = 2𝑜)
9088, 89mto 188 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2𝑜)
9173, 83, 903pm3.2i 1376 . . . . . . . . . . 11 (¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = 2𝑜) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2𝑜))
92 fvex 6314 . . . . . . . . . . . . . 14 ((𝐴𝑋)‘𝑥) ∈ V
9392, 92brtp 31867 . . . . . . . . . . . . 13 (((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴𝑋)‘𝑥) ↔ ((((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = ∅) ∨ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = 2𝑜) ∨ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2𝑜)))
94 3oran 1099 . . . . . . . . . . . . 13 (((((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = ∅) ∨ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = 2𝑜) ∨ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2𝑜)) ↔ ¬ (¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = 2𝑜) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2𝑜)))
9593, 94bitri 264 . . . . . . . . . . . 12 (((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴𝑋)‘𝑥) ↔ ¬ (¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = 2𝑜) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2𝑜)))
9695con2bii 346 . . . . . . . . . . 11 ((¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1𝑜 ∧ ((𝐴𝑋)‘𝑥) = 2𝑜) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2𝑜)) ↔ ¬ ((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴𝑋)‘𝑥))
9791, 96mpbi 220 . . . . . . . . . 10 ¬ ((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴𝑋)‘𝑥)
98 simpl2l 1259 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) → (𝐴𝑋) = (𝐵𝑋))
9998adantr 472 . . . . . . . . . . . 12 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = (𝐵𝑋))
10099fveq1d 6306 . . . . . . . . . . 11 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ((𝐴𝑋)‘𝑥) = ((𝐵𝑋)‘𝑥))
101100breq2d 4772 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴𝑋)‘𝑥) ↔ ((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘𝑥)))
10297, 101mtbii 315 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ ((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘𝑥))
103 fvres 6320 . . . . . . . . . . 11 (𝑥𝑋 → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
104 fvres 6320 . . . . . . . . . . 11 (𝑥𝑋 → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
105103, 104breq12d 4773 . . . . . . . . . 10 (𝑥𝑋 → (((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘𝑥) ↔ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
106105notbid 307 . . . . . . . . 9 (𝑥𝑋 → (¬ ((𝐴𝑋)‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵𝑋)‘𝑥) ↔ ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
107102, 106syl5ibcom 235 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋 → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
10845intnanr 999 . . . . . . . . . . . 12 ¬ (∅ = 1𝑜 ∧ 1𝑜 = ∅)
10945intnanr 999 . . . . . . . . . . . 12 ¬ (∅ = 1𝑜 ∧ 1𝑜 = 2𝑜)
11081intnan 998 . . . . . . . . . . . 12 ¬ (∅ = ∅ ∧ 1𝑜 = 2𝑜)
111108, 109, 1103pm3.2i 1376 . . . . . . . . . . 11 (¬ (∅ = 1𝑜 ∧ 1𝑜 = ∅) ∧ ¬ (∅ = 1𝑜 ∧ 1𝑜 = 2𝑜) ∧ ¬ (∅ = ∅ ∧ 1𝑜 = 2𝑜))
112 0ex 4898 . . . . . . . . . . . . . 14 ∅ ∈ V
113112, 42brtp 31867 . . . . . . . . . . . . 13 (∅{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}1𝑜 ↔ ((∅ = 1𝑜 ∧ 1𝑜 = ∅) ∨ (∅ = 1𝑜 ∧ 1𝑜 = 2𝑜) ∨ (∅ = ∅ ∧ 1𝑜 = 2𝑜)))
114 3oran 1099 . . . . . . . . . . . . 13 (((∅ = 1𝑜 ∧ 1𝑜 = ∅) ∨ (∅ = 1𝑜 ∧ 1𝑜 = 2𝑜) ∨ (∅ = ∅ ∧ 1𝑜 = 2𝑜)) ↔ ¬ (¬ (∅ = 1𝑜 ∧ 1𝑜 = ∅) ∧ ¬ (∅ = 1𝑜 ∧ 1𝑜 = 2𝑜) ∧ ¬ (∅ = ∅ ∧ 1𝑜 = 2𝑜)))
115113, 114bitri 264 . . . . . . . . . . . 12 (∅{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}1𝑜 ↔ ¬ (¬ (∅ = 1𝑜 ∧ 1𝑜 = ∅) ∧ ¬ (∅ = 1𝑜 ∧ 1𝑜 = 2𝑜) ∧ ¬ (∅ = ∅ ∧ 1𝑜 = 2𝑜)))
116115con2bii 346 . . . . . . . . . . 11 ((¬ (∅ = 1𝑜 ∧ 1𝑜 = ∅) ∧ ¬ (∅ = 1𝑜 ∧ 1𝑜 = 2𝑜) ∧ ¬ (∅ = ∅ ∧ 1𝑜 = 2𝑜)) ↔ ¬ ∅{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}1𝑜)
117111, 116mpbi 220 . . . . . . . . . 10 ¬ ∅{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}1𝑜
11846, 47breq12d 4773 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ((𝐴𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑋) ↔ ∅{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}1𝑜))
119117, 118mtbiri 316 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ (𝐴𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑋))
120 fveq2 6304 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
121 fveq2 6304 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
122120, 121breq12d 4773 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑋)))
123122notbid 307 . . . . . . . . 9 (𝑥 = 𝑋 → (¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥) ↔ ¬ (𝐴𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑋)))
124119, 123syl5ibrcom 237 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥 = 𝑋 → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
125107, 124jaod 394 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ((𝑥𝑋𝑥 = 𝑋) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
12670, 125sylbid 230 . . . . . 6 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋 → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
12768, 126mpd 15 . . . . 5 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))
128127expr 644 . . . 4 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
129128ralrimiva 3068 . . 3 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1𝑜) → ∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
13040, 129mtand 694 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ (𝐵𝑋) = 1𝑜)
131 nofv 32037 . . . 4 (𝐵 No → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜 ∨ (𝐵𝑋) = 2𝑜))
13232, 131syl 17 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜 ∨ (𝐵𝑋) = 2𝑜))
133 3orrot 1077 . . . 4 (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜 ∨ (𝐵𝑋) = 2𝑜) ↔ ((𝐵𝑋) = 1𝑜 ∨ (𝐵𝑋) = 2𝑜 ∨ (𝐵𝑋) = ∅))
134 3orrot 1077 . . . 4 (((𝐵𝑋) = 1𝑜 ∨ (𝐵𝑋) = 2𝑜 ∨ (𝐵𝑋) = ∅) ↔ ((𝐵𝑋) = 2𝑜 ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜))
135133, 134bitri 264 . . 3 (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜 ∨ (𝐵𝑋) = 2𝑜) ↔ ((𝐵𝑋) = 2𝑜 ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜))
136132, 135sylib 208 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ((𝐵𝑋) = 2𝑜 ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1𝑜))
13731, 130, 136ecase23d 1549 1 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → (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 1596   ∈ wcel 2103  ∀wral 3014  ∃wrex 3015   ⊆ wss 3680  ∅c0 4023  {ctp 4289  ⟨cop 4291   class class class wbr 4760   Or wor 5138  dom cdm 5218   ↾ cres 5220  Rel wrel 5223  Oncon0 5836  suc csuc 5838  Fun wfun 5995  ‘cfv 6001  1𝑜c1o 7673  2𝑜c2o 7674   No csur 32020
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