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Theorem nogt01o 33542
Description: Given 𝐴 greater than 𝐵, equal to 𝐵 up to 𝑋, and 𝐵(𝑋) undefined, then 𝐴(𝑋) = 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
Assertion
Ref Expression
nogt01o (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = 1o)

Proof of Theorem nogt01o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sltso 33522 . . . 4 <s Or No
2 simp11 1204 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐴 No )
3 sonr 5465 . . . 4 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
41, 2, 3sylancr 590 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ¬ 𝐴 <s 𝐴)
5 simpl2r 1228 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 <s 𝐵)
6 simpl2l 1227 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐴𝑋) = (𝐵𝑋))
7 simpl11 1249 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 No )
8 nofun 33495 . . . . . . . 8 (𝐴 No → Fun 𝐴)
9 funrel 6356 . . . . . . . 8 (Fun 𝐴 → Rel 𝐴)
108, 9syl 17 . . . . . . 7 (𝐴 No → Rel 𝐴)
117, 10syl 17 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → Rel 𝐴)
12 simpl13 1251 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝑋 ∈ On)
13 simpr 488 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐴𝑋) = ∅)
14 nolt02olem 33540 . . . . . . 7 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
157, 12, 13, 14syl3anc 1372 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
16 relssres 5866 . . . . . 6 ((Rel 𝐴 ∧ dom 𝐴𝑋) → (𝐴𝑋) = 𝐴)
1711, 15, 16syl2anc 587 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐴𝑋) = 𝐴)
18 simpl12 1250 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐵 No )
19 nofun 33495 . . . . . . . 8 (𝐵 No → Fun 𝐵)
20 funrel 6356 . . . . . . . 8 (Fun 𝐵 → Rel 𝐵)
2119, 20syl 17 . . . . . . 7 (𝐵 No → Rel 𝐵)
2218, 21syl 17 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → Rel 𝐵)
23 simpl3 1194 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐵𝑋) = ∅)
24 nolt02olem 33540 . . . . . . 7 ((𝐵 No 𝑋 ∈ On ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
2518, 12, 23, 24syl3anc 1372 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → dom 𝐵𝑋)
26 relssres 5866 . . . . . 6 ((Rel 𝐵 ∧ dom 𝐵𝑋) → (𝐵𝑋) = 𝐵)
2722, 25, 26syl2anc 587 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐵𝑋) = 𝐵)
286, 17, 273eqtr3d 2781 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 = 𝐵)
295, 28breqtrrd 5058 . . 3 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 <s 𝐴)
304, 29mtand 816 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ¬ (𝐴𝑋) = ∅)
31 simp2r 1201 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐴 <s 𝐵)
32 simp12 1205 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐵 No )
33 sltval 33493 . . . . . 6 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
342, 32, 33syl2anc 587 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
3531, 34mpbid 235 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
36 ralinexa 3174 . . . . 5 (∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ ¬ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
3736con2bii 361 . . . 4 (∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ ¬ ∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
3835, 37sylib 221 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ¬ ∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
39 1n0 8150 . . . . . . . . . . . 12 1o ≠ ∅
4039neii 2936 . . . . . . . . . . 11 ¬ 1o = ∅
41 eqtr2 2759 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) → 1o = ∅)
4240, 41mto 200 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅)
43 df-2o 8132 . . . . . . . . . . . . 13 2o = suc 1o
44 2on 8139 . . . . . . . . . . . . . . . 16 2o ∈ On
4543, 44eqeltrri 2830 . . . . . . . . . . . . . . 15 suc 1o ∈ On
4645onordi 6277 . . . . . . . . . . . . . 14 Ord suc 1o
47 1oex 8144 . . . . . . . . . . . . . . 15 1o ∈ V
4847sucid 6251 . . . . . . . . . . . . . 14 1o ∈ suc 1o
49 nordeq 6191 . . . . . . . . . . . . . 14 ((Ord suc 1o ∧ 1o ∈ suc 1o) → suc 1o ≠ 1o)
5046, 48, 49mp2an 692 . . . . . . . . . . . . 13 suc 1o ≠ 1o
5143, 50eqnetri 3004 . . . . . . . . . . . 12 2o ≠ 1o
5251nesymi 2991 . . . . . . . . . . 11 ¬ 1o = 2o
53 eqtr2 2759 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) → 1o = 2o)
5452, 53mto 200 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o)
55 2on0 8140 . . . . . . . . . . . 12 2o ≠ ∅
5655nesymi 2991 . . . . . . . . . . 11 ¬ ∅ = 2o
57 eqtr2 2759 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o) → ∅ = 2o)
5856, 57mto 200 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o)
5942, 54, 583pm3.2i 1340 . . . . . . . . 9 (¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o))
60 fvex 6687 . . . . . . . . . . . 12 ((𝐴𝑋)‘𝑥) ∈ V
6160, 60brtp 33290 . . . . . . . . . . 11 (((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥) ↔ ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) ∨ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) ∨ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o)))
62 3oran 1110 . . . . . . . . . . 11 (((((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)))
6361, 62bitri 278 . . . . . . . . . 10 (((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥) ↔ ¬ (¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o)))
6463con2bii 361 . . . . . . . . 9 ((¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o)) ↔ ¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥))
6559, 64mpbi 233 . . . . . . . 8 ¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥)
66 simpl2l 1227 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) → (𝐴𝑋) = (𝐵𝑋))
6766adantr 484 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = (𝐵𝑋))
6867fveq1d 6676 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ((𝐴𝑋)‘𝑥) = ((𝐵𝑋)‘𝑥))
6968breq2d 5042 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥) ↔ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥)))
7065, 69mtbii 329 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥))
71 fvres 6693 . . . . . . . . 9 (𝑥𝑋 → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
72 fvres 6693 . . . . . . . . 9 (𝑥𝑋 → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
7371, 72breq12d 5043 . . . . . . . 8 (𝑥𝑋 → (((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
7473notbid 321 . . . . . . 7 (𝑥𝑋 → (¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
7570, 74syl5ibcom 248 . . . . . 6 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋 → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
7651neii 2936 . . . . . . . . . . 11 ¬ 2o = 1o
7776intnanr 491 . . . . . . . . . 10 ¬ (2o = 1o ∧ ∅ = ∅)
7856intnan 490 . . . . . . . . . 10 ¬ (2o = 1o ∧ ∅ = 2o)
7956intnan 490 . . . . . . . . . 10 ¬ (2o = ∅ ∧ ∅ = 2o)
8077, 78, 793pm3.2i 1340 . . . . . . . . 9 (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o))
81 2oex 8148 . . . . . . . . . . . 12 2o ∈ V
82 0ex 5175 . . . . . . . . . . . 12 ∅ ∈ V
8381, 82brtp 33290 . . . . . . . . . . 11 (2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅ ↔ ((2o = 1o ∧ ∅ = ∅) ∨ (2o = 1o ∧ ∅ = 2o) ∨ (2o = ∅ ∧ ∅ = 2o)))
84 3oran 1110 . . . . . . . . . . 11 (((2o = 1o ∧ ∅ = ∅) ∨ (2o = 1o ∧ ∅ = 2o) ∨ (2o = ∅ ∧ ∅ = 2o)) ↔ ¬ (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)))
8583, 84bitri 278 . . . . . . . . . 10 (2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅ ↔ ¬ (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)))
8685con2bii 361 . . . . . . . . 9 ((¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)) ↔ ¬ 2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅)
8780, 86mpbi 233 . . . . . . . 8 ¬ 2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅
88 simplr 769 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = 2o)
89 simpll3 1215 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐵𝑋) = ∅)
9088, 89breq12d 5043 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ((𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋) ↔ 2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅))
9187, 90mtbiri 330 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋))
92 fveq2 6674 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
93 fveq2 6674 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
9492, 93breq12d 5043 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
9594notbid 321 . . . . . . 7 (𝑥 = 𝑋 → (¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ ¬ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
9691, 95syl5ibrcom 250 . . . . . 6 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥 = 𝑋 → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
97 fveq2 6674 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
98 fveq2 6674 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝐵𝑦) = (𝐵𝑋))
9997, 98eqeq12d 2754 . . . . . . . . . . . 12 (𝑦 = 𝑋 → ((𝐴𝑦) = (𝐵𝑦) ↔ (𝐴𝑋) = (𝐵𝑋)))
10099rspccv 3523 . . . . . . . . . . 11 (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → (𝑋𝑥 → (𝐴𝑋) = (𝐵𝑋)))
101100ad2antll 729 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑋𝑥 → (𝐴𝑋) = (𝐵𝑋)))
102 eqcom 2745 . . . . . . . . . 10 ((𝐴𝑋) = (𝐵𝑋) ↔ (𝐵𝑋) = (𝐴𝑋))
103101, 102syl6ib 254 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑋𝑥 → (𝐵𝑋) = (𝐴𝑋)))
10489, 88eqeq12d 2754 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ((𝐵𝑋) = (𝐴𝑋) ↔ ∅ = 2o))
105103, 104sylibd 242 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑋𝑥 → ∅ = 2o))
10656, 105mtoi 202 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ 𝑋𝑥)
107 simprl 771 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑥 ∈ On)
108 simpl13 1251 . . . . . . . . 9 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) → 𝑋 ∈ On)
109108adantr 484 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑋 ∈ On)
110 ontri1 6206 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
111 onsseleq 6213 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
112110, 111bitr3d 284 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (¬ 𝑋𝑥 ↔ (𝑥𝑋𝑥 = 𝑋)))
113107, 109, 112syl2anc 587 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (¬ 𝑋𝑥 ↔ (𝑥𝑋𝑥 = 𝑋)))
114106, 113mpbid 235 . . . . . 6 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑥𝑋𝑥 = 𝑋))
11575, 96, 114mpjaod 859 . . . . 5 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))
116115expr 460 . . . 4 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
117116ralrimiva 3096 . . 3 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) → ∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
11838, 117mtand 816 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ¬ (𝐴𝑋) = 2o)
119 nofv 33503 . . . 4 (𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o))
1202, 119syl 17 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o))
121 3orcoma 1094 . . 3 (((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o) ↔ ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 2o))
122120, 121sylib 221 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 2o))
12330, 118, 122ecase23d 1474 1 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → (𝐴𝑋) = 1o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846  w3o 1087  w3a 1088   = wceq 1542  wcel 2114  wne 2934  wral 3053  wrex 3054  wss 3843  c0 4211  {ctp 4520  cop 4522   class class class wbr 5030   Or wor 5441  dom cdm 5525  cres 5527  Rel wrel 5530  Ord word 6171  Oncon0 6172  suc csuc 6174  Fun wfun 6333  cfv 6339  1oc1o 8124  2oc2o 8125   No csur 33486   <s cslt 33487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-1o 8131  df-2o 8132  df-no 33489  df-slt 33490
This theorem is referenced by:  noinfbnd1lem4  33572
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