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Theorem nolt02o 33096
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 1195 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐴 No )
2 sltso 33078 . . . . . 6 <s Or No
3 sonr 5489 . . . . . 6 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
42, 3mpan 686 . . . . 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 1192 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐴 <s 𝐵)
7 breq2 5061 . . . . 5 (𝐴 = 𝐵 → (𝐴 <s 𝐴𝐴 <s 𝐵))
86, 7syl5ibrcom 248 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → (𝐴 = 𝐵𝐴 <s 𝐴))
95, 8mtod 199 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ 𝐴 = 𝐵)
10 simpl2l 1218 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = (𝐵𝑋))
11 simpl11 1240 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐴 No )
12 nofun 33053 . . . . . 6 (𝐴 No → Fun 𝐴)
13 funrel 6365 . . . . . 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 1242 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝑋 ∈ On)
16 simpl3 1185 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = ∅)
17 nolt02olem 33095 . . . . . 6 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
1811, 15, 16, 17syl3anc 1363 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → dom 𝐴𝑋)
19 relssres 5886 . . . . 5 ((Rel 𝐴 ∧ dom 𝐴𝑋) → (𝐴𝑋) = 𝐴)
2014, 18, 19syl2anc 584 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = 𝐴)
21 simpl12 1241 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐵 No )
22 nofun 33053 . . . . . 6 (𝐵 No → Fun 𝐵)
23 funrel 6365 . . . . . 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 485 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐵𝑋) = ∅)
26 nolt02olem 33095 . . . . . 6 ((𝐵 No 𝑋 ∈ On ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
2721, 15, 25, 26syl3anc 1363 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
28 relssres 5886 . . . . 5 ((Rel 𝐵 ∧ dom 𝐵𝑋) → (𝐵𝑋) = 𝐵)
2924, 27, 28syl2anc 584 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → (𝐵𝑋) = 𝐵)
3010, 20, 293eqtr3d 2861 . . 3 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = ∅) → 𝐴 = 𝐵)
319, 30mtand 812 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ (𝐵𝑋) = ∅)
32 simp12 1196 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → 𝐵 No )
33 sltval 33051 . . . . . 6 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
341, 32, 33syl2anc 584 . . . . 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 233 . . . 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 397 . . . . . 6 ((∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ ¬ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
3736rexbii 3244 . . . . 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 3235 . . . . 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 276 . . . 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 219 . . 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 8099 . . . . . . . . . . . 12 1o ∈ V
4241prid1 4690 . . . . . . . . . . 11 1o ∈ {1o, 2o}
4342nosgnn0i 33063 . . . . . . . . . 10 ∅ ≠ 1o
4443neii 3015 . . . . . . . . 9 ¬ ∅ = 1o
45 simpll3 1206 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = ∅)
46 simplr 765 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐵𝑋) = 1o)
47 eqeq1 2822 . . . . . . . . . . . . 13 ((𝐴𝑋) = (𝐵𝑋) → ((𝐴𝑋) = ∅ ↔ (𝐵𝑋) = ∅))
4847anbi1d 629 . . . . . . . . . . . 12 ((𝐴𝑋) = (𝐵𝑋) → (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1o) ↔ ((𝐵𝑋) = ∅ ∧ (𝐵𝑋) = 1o)))
49 eqtr2 2839 . . . . . . . . . . . 12 (((𝐵𝑋) = ∅ ∧ (𝐵𝑋) = 1o) → ∅ = 1o)
5048, 49syl6bi 254 . . . . . . . . . . 11 ((𝐴𝑋) = (𝐵𝑋) → (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1o) → ∅ = 1o))
5150com12 32 . . . . . . . . . 10 (((𝐴𝑋) = ∅ ∧ (𝐵𝑋) = 1o) → ((𝐴𝑋) = (𝐵𝑋) → ∅ = 1o))
5245, 46, 51syl2anc 584 . . . . . . . . 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 200 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ (𝐴𝑋) = (𝐵𝑋))
54 simpr 485 . . . . . . . . 9 ((((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) ∧ 𝑋𝑥) → 𝑋𝑥)
55 simplrr 774 . . . . . . . . 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 6663 . . . . . . . . . . 11 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
57 fveq2 6663 . . . . . . . . . . 11 (𝑦 = 𝑋 → (𝐵𝑦) = (𝐵𝑋))
5856, 57eqeq12d 2834 . . . . . . . . . 10 (𝑦 = 𝑋 → ((𝐴𝑦) = (𝐵𝑦) ↔ (𝐴𝑋) = (𝐵𝑋)))
5958rspcv 3615 . . . . . . . . 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 812 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ 𝑋𝑥)
62 simprl 767 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑥 ∈ On)
63 simpl13 1242 . . . . . . . . 9 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) → 𝑋 ∈ On)
6463adantr 481 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑋 ∈ On)
65 ontri1 6218 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
6662, 64, 65syl2anc 584 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
6761, 66mpbird 258 . . . . . 6 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑥𝑋)
68 onsseleq 6225 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
6962, 64, 68syl2anc 584 . . . . . . 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 2839 . . . . . . . . . . . . . 14 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 1o) → ∅ = 1o)
7170ancoms 459 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) → ∅ = 1o)
7244, 71mto 198 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅)
73 df-1o 8091 . . . . . . . . . . . . . . . 16 1o = suc ∅
74 df-2o 8092 . . . . . . . . . . . . . . . 16 2o = suc 1o
7573, 74eqeq12i 2833 . . . . . . . . . . . . . . 15 (1o = 2o ↔ suc ∅ = suc 1o)
76 0elon 6237 . . . . . . . . . . . . . . . 16 ∅ ∈ On
77 1on 8098 . . . . . . . . . . . . . . . 16 1o ∈ On
78 suc11 6287 . . . . . . . . . . . . . . . 16 ((∅ ∈ On ∧ 1o ∈ On) → (suc ∅ = suc 1o ↔ ∅ = 1o))
7976, 77, 78mp2an 688 . . . . . . . . . . . . . . 15 (suc ∅ = suc 1o ↔ ∅ = 1o)
8075, 79bitri 276 . . . . . . . . . . . . . 14 (1o = 2o ↔ ∅ = 1o)
8143, 80nemtbir 3109 . . . . . . . . . . . . 13 ¬ 1o = 2o
82 eqtr2 2839 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) → 1o = 2o)
8381, 82mto 198 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o)
84 2on 8100 . . . . . . . . . . . . . . . . 17 2o ∈ On
8584elexi 3511 . . . . . . . . . . . . . . . 16 2o ∈ V
8685prid2 4691 . . . . . . . . . . . . . . 15 2o ∈ {1o, 2o}
8786nosgnn0i 33063 . . . . . . . . . . . . . 14 ∅ ≠ 2o
8887neii 3015 . . . . . . . . . . . . 13 ¬ ∅ = 2o
89 eqtr2 2839 . . . . . . . . . . . . 13 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o) → ∅ = 2o)
9088, 89mto 198 . . . . . . . . . . . 12 ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o)
9172, 83, 903pm3.2i 1331 . . . . . . . . . . 11 (¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o))
92 fvex 6676 . . . . . . . . . . . . . 14 ((𝐴𝑋)‘𝑥) ∈ V
9392, 92brtp 32882 . . . . . . . . . . . . 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 1101 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 359 . . . . . . . . . . 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 231 . . . . . . . . . 10 ¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥)
98 simpl2l 1218 . . . . . . . . . . . . 13 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) → (𝐴𝑋) = (𝐵𝑋))
9998adantr 481 . . . . . . . . . . . 12 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) ∧ (𝐵𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = (𝐵𝑋))
10099fveq1d 6665 . . . . . . . . . . 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 5069 . . . . . . . . . 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 327 . . . . . . . . 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 6682 . . . . . . . . . . 11 (𝑥𝑋 → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
104 fvres 6682 . . . . . . . . . . 11 (𝑥𝑋 → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
105103, 104breq12d 5070 . . . . . . . . . 10 (𝑥𝑋 → (((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
106105notbid 319 . . . . . . . . 9 (𝑥𝑋 → (¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
107102, 106syl5ibcom 246 . . . . . . . 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 488 . . . . . . . . . . . 12 ¬ (∅ = 1o ∧ 1o = ∅)
10944intnanr 488 . . . . . . . . . . . 12 ¬ (∅ = 1o ∧ 1o = 2o)
11081intnan 487 . . . . . . . . . . . 12 ¬ (∅ = ∅ ∧ 1o = 2o)
111108, 109, 1103pm3.2i 1331 . . . . . . . . . . 11 (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o))
112 0ex 5202 . . . . . . . . . . . . . 14 ∅ ∈ V
113112, 41brtp 32882 . . . . . . . . . . . . 13 (∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o ↔ ((∅ = 1o ∧ 1o = ∅) ∨ (∅ = 1o ∧ 1o = 2o) ∨ (∅ = ∅ ∧ 1o = 2o)))
114 3oran 1101 . . . . . . . . . . . . 13 (((∅ = 1o ∧ 1o = ∅) ∨ (∅ = 1o ∧ 1o = 2o) ∨ (∅ = ∅ ∧ 1o = 2o)) ↔ ¬ (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)))
115113, 114bitri 276 . . . . . . . . . . . 12 (∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o ↔ ¬ (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)))
116115con2bii 359 . . . . . . . . . . 11 ((¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)) ↔ ¬ ∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o)
117111, 116mpbi 231 . . . . . . . . . 10 ¬ ∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o
11845, 46breq12d 5070 . . . . . . . . . 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 328 . . . . . . . . 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 6663 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
121 fveq2 6663 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
122120, 121breq12d 5070 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
123122notbid 319 . . . . . . . . 9 (𝑥 = 𝑋 → (¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ ¬ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
124119, 123syl5ibrcom 248 . . . . . . . 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 853 . . . . . . 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 241 . . . . . 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 457 . . . 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 3179 . . 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 812 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ¬ (𝐵𝑋) = 1o)
131 nofv 33061 . . . 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 1084 . . . 4 (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o) ↔ ((𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅))
134 3orrot 1084 . . . 4 (((𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅) ↔ ((𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o))
135133, 134bitri 276 . . 3 (((𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o ∨ (𝐵𝑋) = 2o) ↔ ((𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o))
136132, 135sylib 219 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴𝑋) = ∅) → ((𝐵𝑋) = 2o ∨ (𝐵𝑋) = ∅ ∨ (𝐵𝑋) = 1o))
13731, 130, 136ecase23d 1464 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 207  wa 396  wo 841  w3o 1078  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933  c0 4288  {ctp 4561  cop 4563   class class class wbr 5057   Or wor 5466  dom cdm 5548  cres 5550  Rel wrel 5553  Oncon0 6184  suc csuc 6186  Fun wfun 6342  cfv 6348  1oc1o 8084  2oc2o 8085   No csur 33044   <s cslt 33045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-1o 8091  df-2o 8092  df-no 33047  df-slt 33048
This theorem is referenced by:  nosupbnd1lem4  33108
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