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Theorem nolt02o 27066
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 1204 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐴 No )
2 sltso 27047 . . . . . 6 <s Or No
3 sonr 5572 . . . . . 6 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
42, 3mpan 689 . . . . 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 1201 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐴 <s 𝐵)
7 breq2 5113 . . . . 5 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
86, 7syl5ibrcom 247 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → (𝐴 = 𝐵𝐴 <s 𝐴))
95, 8mtod 197 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ 𝐴 = 𝐵)
10 simpl2l 1227 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = (𝐵𝑋))
11 simpl11 1249 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐴 No )
12 nofun 27020 . . . . . 6 (𝐴 No → Fun 𝐴)
13 funrel 6522 . . . . . 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 1251 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝑋 ∈ On)
16 simpl3 1194 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = ∅)
17 nolt02olem 27065 . . . . . 6 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
1811, 15, 16, 17syl3anc 1372 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → dom 𝐴𝑋)
19 relssres 5982 . . . . 5 ((Rel 𝐴 ∧ dom 𝐴𝑋) → (𝐴𝑋) = 𝐴)
2014, 18, 19syl2anc 585 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = 𝐴)
21 simpl12 1250 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐵 No )
22 nofun 27020 . . . . . 6 (𝐵 No → Fun 𝐵)
23 funrel 6522 . . . . . 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 486 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐵𝑋) = ∅)
26 nolt02olem 27065 . . . . . 6 ((𝐵 No 𝑋 ∈ On ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
2721, 15, 25, 26syl3anc 1372 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
28 relssres 5982 . . . . 5 ((Rel 𝐵 ∧ dom 𝐵𝑋) → (𝐵𝑋) = 𝐵)
2924, 27, 28syl2anc 585 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐵𝑋) = 𝐵)
3010, 20, 293eqtr3d 2781 . . 3 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐴 = 𝐵)
319, 30mtand 815 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ (𝐵𝑋) = ∅)
32 simp12 1205 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐵 No )
33 sltval 27018 . . . . . 6 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
341, 32, 33syl2anc 585 . . . . 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 231 . . . 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 398 . . . . . 6 ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ ¬ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
3736rexbii 3094 . . . . 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 3100 . . . . 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 275 . . . 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 217 . . 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 1oex 8426 . . . . . . . . . . . 12 1o ∈ V
4241prid1 4727 . . . . . . . . . . 11 1o ∈ {1o, 2o}
4342nosgnn0i 27030 . . . . . . . . . 10 ∅ ≠ 1o
4443neii 2942 . . . . . . . . 9 ¬ ∅ = 1o
45 simpll3 1215 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = ∅)
46 simplr 768 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐵𝑋) = 1o)
47 eqeq1 2737 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋) = ∅ ↔ (𝐵𝑋) = ∅))
4847anbi1d 631 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1o) ↔ ((𝐵𝑋) = ∅ ∧ (𝐵𝑋) = 1o)))
49 eqtr2 2757 . . . . . . . . . . . 12 (((𝐵𝑋) = ∅ ∧ (𝐵𝑋) = 1o) → ∅ = 1o)
5048, 49syl6bi 253 . . . . . . . . . . 11 ((𝐴𝑋) = (𝐵𝑋) → (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1o) → ∅ = 1o))
5150com12 32 . . . . . . . . . 10 (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1o) → ((𝐴𝑋) = (𝐵𝑋) → ∅ = 1o))
5245, 46, 51syl2anc 585 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ((𝐴𝑋) = (𝐵𝑋) → ∅ = 1o))
5344, 52mtoi 198 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ (𝐴𝑋) = (𝐵𝑋))
54 simpr 486 . . . . . . . . 9 ((((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) ∧ 𝑋𝑥) → 𝑋𝑥)
55 simplrr 777 . . . . . . . . 9 ((((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) ∧ 𝑋𝑥) → ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))
56 fveq2 6846 . . . . . . . . . . 11 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
57 fveq2 6846 . . . . . . . . . . 11 (𝑦 = 𝑋 → (𝐵𝑦) = (𝐵𝑋))
5856, 57eqeq12d 2749 . . . . . . . . . 10 (𝑦 = 𝑋 → ((𝐴𝑦) = (𝐵𝑦) ↔ (𝐴𝑋) = (𝐵𝑋)))
5958rspcv 3579 . . . . . . . . 9 (𝑋𝑥 → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → (𝐴𝑋) = (𝐵𝑋)))
6054, 55, 59sylc 65 . . . . . . . 8 ((((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) ∧ 𝑋𝑥) → (𝐴𝑋) = (𝐵𝑋))
6153, 60mtand 815 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ 𝑋𝑥)
62 simprl 770 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑥 ∈ On)
63 simpl13 1251 . . . . . . . . 9 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) → 𝑋 ∈ On)
6463adantr 482 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑋 ∈ On)
65 ontri1 6355 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
6662, 64, 65syl2anc 585 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
6761, 66mpbird 257 . . . . . 6 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑥𝑋)
68 onsseleq 6362 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
6962, 64, 68syl2anc 585 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
70 eqtr2 2757 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 1o) → ∅ = 1o)
7170ancoms 460 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) → ∅ = 1o)
7244, 71mto 196 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅)
73 df-1o 8416 . . . . . . . . . . . . . . . 16 1o = suc ∅
74 df-2o 8417 . . . . . . . . . . . . . . . 16 2o = suc 1o
7573, 74eqeq12i 2751 . . . . . . . . . . . . . . 15 (1o = 2o ↔ suc ∅ = suc 1o)
76 0elon 6375 . . . . . . . . . . . . . . . 16 ∅ ∈ On
77 1on 8428 . . . . . . . . . . . . . . . 16 1o ∈ On
78 suc11 6428 . . . . . . . . . . . . . . . 16 ((∅ ∈ On ∧ 1o ∈ On) → (suc ∅ = suc 1o ↔ ∅ = 1o))
7976, 77, 78mp2an 691 . . . . . . . . . . . . . . 15 (suc ∅ = suc 1o ↔ ∅ = 1o)
8075, 79bitri 275 . . . . . . . . . . . . . 14 (1o = 2o ↔ ∅ = 1o)
8143, 80nemtbir 3037 . . . . . . . . . . . . 13 ¬ 1o = 2o
82 eqtr2 2757 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) → 1o = 2o)
8381, 82mto 196 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o)
84 2on 8430 . . . . . . . . . . . . . . . . 17 2o ∈ On
8584elexi 3466 . . . . . . . . . . . . . . . 16 2o ∈ V
8685prid2 4728 . . . . . . . . . . . . . . 15 2o ∈ {1o, 2o}
8786nosgnn0i 27030 . . . . . . . . . . . . . 14 ∅ ≠ 2o
8887neii 2942 . . . . . . . . . . . . 13 ¬ ∅ = 2o
89 eqtr2 2757 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o) → ∅ = 2o)
9088, 89mto 196 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o)
9172, 83, 903pm3.2i 1340 . . . . . . . . . . 11 (¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o))
92 fvex 6859 . . . . . . . . . . . . . 14 ((𝐴𝑋)‘𝑥) ∈ V
9392, 92brtp 5484 . . . . . . . . . . . . 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 1110 . . . . . . . . . . . . 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 275 . . . . . . . . . . . 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 358 . . . . . . . . . . 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 229 . . . . . . . . . 10 ¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥)
98 simpl2l 1227 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) → (𝐴𝑋) = (𝐵𝑋))
9998adantr 482 . . . . . . . . . . . 12 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = (𝐵𝑋))
10099fveq1d 6848 . . . . . . . . . . 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 5121 . . . . . . . . . 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 326 . . . . . . . . 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 6865 . . . . . . . . . . 11 (𝑥𝑋 → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
104 fvres 6865 . . . . . . . . . . 11 (𝑥𝑋 → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
105103, 104breq12d 5122 . . . . . . . . . 10 (𝑥𝑋 → (((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
106105notbid 318 . . . . . . . . 9 (𝑥𝑋 → (¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
107102, 106syl5ibcom 244 . . . . . . . 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)))
10844intnanr 489 . . . . . . . . . . . 12 ¬ (∅ = 1o ∧ 1o = ∅)
10944intnanr 489 . . . . . . . . . . . 12 ¬ (∅ = 1o ∧ 1o = 2o)
11081intnan 488 . . . . . . . . . . . 12 ¬ (∅ = ∅ ∧ 1o = 2o)
111108, 109, 1103pm3.2i 1340 . . . . . . . . . . 11 (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o))
112 0ex 5268 . . . . . . . . . . . . . 14 ∅ ∈ V
113112, 41brtp 5484 . . . . . . . . . . . . 13 (∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o ↔ ((∅ = 1o ∧ 1o = ∅) ∨ (∅ = 1o ∧ 1o = 2o) ∨ (∅ = ∅ ∧ 1o = 2o)))
114 3oran 1110 . . . . . . . . . . . . 13 (((∅ = 1o ∧ 1o = ∅) ∨ (∅ = 1o ∧ 1o = 2o) ∨ (∅ = ∅ ∧ 1o = 2o)) ↔ ¬ (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)))
115113, 114bitri 275 . . . . . . . . . . . 12 (∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o ↔ ¬ (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)))
116115con2bii 358 . . . . . . . . . . 11 ((¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)) ↔ ¬ ∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o)
117111, 116mpbi 229 . . . . . . . . . 10 ¬ ∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o
11845, 46breq12d 5122 . . . . . . . . . 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 327 . . . . . . . . 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 6846 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
121 fveq2 6846 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
122120, 121breq12d 5122 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
123122notbid 318 . . . . . . . . 9 (𝑥 = 𝑋 → (¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ ¬ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
124119, 123syl5ibrcom 247 . . . . . . . 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 858 . . . . . . 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)))
12669, 125sylbid 239 . . . . . 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)))
12767, 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 458 . . . 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 3140 . . 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 815 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ (𝐵𝑋) = 1o)
131 nofv 27028 . . . 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 1093 . . . 4 (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o) ↔ ((𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅))
134 3orrot 1093 . . . 4 (((𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅) ↔ ((𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o))
135133, 134bitri 275 . . 3 (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o) ↔ ((𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o))
136132, 135sylib 217 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ((𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o))
13731, 130, 136ecase23d 1474 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 205  wa 397  wo 846  w3o 1087  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  wss 3914  c0 4286  {ctp 4594  cop 4596   class class class wbr 5109   Or wor 5548  dom cdm 5637  cres 5639  Rel wrel 5642  Oncon0 6321  suc csuc 6323  Fun wfun 6494  cfv 6500  1oc1o 8409  2oc2o 8410   No csur 27011   <s cslt 27012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1o 8416  df-2o 8417  df-no 27014  df-slt 27015
This theorem is referenced by:  nosupbnd1lem4  27082
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