MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nogt01o Structured version   Visualization version   GIF version

Theorem nogt01o 27756
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 27736 . . . 4 <s Or No
2 simp11 1202 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐴 No )
3 sonr 5621 . . . 4 (( <s Or No 𝐴 No ) → ¬ 𝐴 <s 𝐴)
41, 2, 3sylancr 587 . . 3 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ¬ 𝐴 <s 𝐴)
5 simpl2r 1226 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 <s 𝐵)
6 simpl2l 1225 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐴𝑋) = (𝐵𝑋))
7 simpl11 1247 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 No )
8 nofun 27709 . . . . . . . 8 (𝐴 No → Fun 𝐴)
9 funrel 6585 . . . . . . . 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 1249 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝑋 ∈ On)
13 simpr 484 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐴𝑋) = ∅)
14 nolt02olem 27754 . . . . . . 7 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
157, 12, 13, 14syl3anc 1370 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
16 relssres 6042 . . . . . 6 ((Rel 𝐴 ∧ dom 𝐴𝑋) → (𝐴𝑋) = 𝐴)
1711, 15, 16syl2anc 584 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐴𝑋) = 𝐴)
18 simpl12 1248 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐵 No )
19 nofun 27709 . . . . . . . 8 (𝐵 No → Fun 𝐵)
20 funrel 6585 . . . . . . . 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 1192 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐵𝑋) = ∅)
24 nolt02olem 27754 . . . . . . 7 ((𝐵 No 𝑋 ∈ On ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
2518, 12, 23, 24syl3anc 1370 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → dom 𝐵𝑋)
26 relssres 6042 . . . . . 6 ((Rel 𝐵 ∧ dom 𝐵𝑋) → (𝐵𝑋) = 𝐵)
2722, 25, 26syl2anc 584 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐵𝑋) = 𝐵)
286, 17, 273eqtr3d 2783 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 = 𝐵)
295, 28breqtrrd 5176 . . 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 1199 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐴 <s 𝐵)
32 simp12 1203 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐵 No )
33 sltval 27707 . . . . . 6 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
342, 32, 33syl2anc 584 . . . . 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 232 . . . 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 3099 . . . . 5 (∀𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)) ↔ ¬ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
3736con2bii 357 . . . 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 218 . . 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 8525 . . . . . . . . . . . 12 1o ≠ ∅
4039neii 2940 . . . . . . . . . . 11 ¬ 1o = ∅
41 eqtr2 2759 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) → 1o = ∅)
4240, 41mto 197 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅)
43 df-2o 8506 . . . . . . . . . . . . 13 2o = suc 1o
44 2on 8519 . . . . . . . . . . . . . . . 16 2o ∈ On
4543, 44eqeltrri 2836 . . . . . . . . . . . . . . 15 suc 1o ∈ On
4645onordi 6497 . . . . . . . . . . . . . 14 Ord suc 1o
47 1oex 8515 . . . . . . . . . . . . . . 15 1o ∈ V
4847sucid 6468 . . . . . . . . . . . . . 14 1o ∈ suc 1o
49 nordeq 6405 . . . . . . . . . . . . . 14 ((Ord suc 1o ∧ 1o ∈ suc 1o) → suc 1o ≠ 1o)
5046, 48, 49mp2an 692 . . . . . . . . . . . . 13 suc 1o ≠ 1o
5143, 50eqnetri 3009 . . . . . . . . . . . 12 2o ≠ 1o
5251nesymi 2996 . . . . . . . . . . 11 ¬ 1o = 2o
53 eqtr2 2759 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) → 1o = 2o)
5452, 53mto 197 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o)
55 2on0 8521 . . . . . . . . . . . 12 2o ≠ ∅
5655nesymi 2996 . . . . . . . . . . 11 ¬ ∅ = 2o
57 eqtr2 2759 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o) → ∅ = 2o)
5856, 57mto 197 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o)
5942, 54, 583pm3.2i 1338 . . . . . . . . 9 (¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o))
60 fvex 6920 . . . . . . . . . . . 12 ((𝐴𝑋)‘𝑥) ∈ V
6160, 60brtp 5533 . . . . . . . . . . 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 1108 . . . . . . . . . . 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 275 . . . . . . . . . 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 357 . . . . . . . . 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 230 . . . . . . . 8 ¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥)
66 simpl2l 1225 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) → (𝐴𝑋) = (𝐵𝑋))
6766adantr 480 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = (𝐵𝑋))
6867fveq1d 6909 . . . . . . . . 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 5160 . . . . . . . 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 326 . . . . . . 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 6926 . . . . . . . . 9 (𝑥𝑋 → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
72 fvres 6926 . . . . . . . . 9 (𝑥𝑋 → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
7371, 72breq12d 5161 . . . . . . . 8 (𝑥𝑋 → (((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
7473notbid 318 . . . . . . 7 (𝑥𝑋 → (¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
7570, 74syl5ibcom 245 . . . . . 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 2940 . . . . . . . . . . 11 ¬ 2o = 1o
7776intnanr 487 . . . . . . . . . 10 ¬ (2o = 1o ∧ ∅ = ∅)
7856intnan 486 . . . . . . . . . 10 ¬ (2o = 1o ∧ ∅ = 2o)
7956intnan 486 . . . . . . . . . 10 ¬ (2o = ∅ ∧ ∅ = 2o)
8077, 78, 793pm3.2i 1338 . . . . . . . . 9 (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o))
81 2oex 8516 . . . . . . . . . . . 12 2o ∈ V
82 0ex 5313 . . . . . . . . . . . 12 ∅ ∈ V
8381, 82brtp 5533 . . . . . . . . . . 11 (2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅ ↔ ((2o = 1o ∧ ∅ = ∅) ∨ (2o = 1o ∧ ∅ = 2o) ∨ (2o = ∅ ∧ ∅ = 2o)))
84 3oran 1108 . . . . . . . . . . 11 (((2o = 1o ∧ ∅ = ∅) ∨ (2o = 1o ∧ ∅ = 2o) ∨ (2o = ∅ ∧ ∅ = 2o)) ↔ ¬ (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)))
8583, 84bitri 275 . . . . . . . . . 10 (2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅ ↔ ¬ (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)))
8685con2bii 357 . . . . . . . . 9 ((¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)) ↔ ¬ 2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅)
8780, 86mpbi 230 . . . . . . . 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 1213 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐵𝑋) = ∅)
9088, 89breq12d 5161 . . . . . . . 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 327 . . . . . . 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 6907 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
93 fveq2 6907 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
9492, 93breq12d 5161 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
9594notbid 318 . . . . . . 7 (𝑥 = 𝑋 → (¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ ¬ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
9691, 95syl5ibrcom 247 . . . . . 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 6907 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
98 fveq2 6907 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝐵𝑦) = (𝐵𝑋))
9997, 98eqeq12d 2751 . . . . . . . . . . . 12 (𝑦 = 𝑋 → ((𝐴𝑦) = (𝐵𝑦) ↔ (𝐴𝑋) = (𝐵𝑋)))
10099rspccv 3619 . . . . . . . . . . 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 2742 . . . . . . . . . 10 ((𝐴𝑋) = (𝐵𝑋) ↔ (𝐵𝑋) = (𝐴𝑋))
103101, 102imbitrdi 251 . . . . . . . . 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 2751 . . . . . . . . 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 239 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑋𝑥 → ∅ = 2o))
10656, 105mtoi 199 . . . . . . 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 1249 . . . . . . . . 9 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) → 𝑋 ∈ On)
109108adantr 480 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑋 ∈ On)
110 ontri1 6420 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
111 onsseleq 6427 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
112110, 111bitr3d 281 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (¬ 𝑋𝑥 ↔ (𝑥𝑋𝑥 = 𝑋)))
113107, 109, 112syl2anc 584 . . . . . . 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 232 . . . . . 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 860 . . . . 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 456 . . . 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 3144 . . 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 27717 . . . 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 1092 . . 3 (((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o) ↔ ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 2o))
122120, 121sylib 218 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 2o))
12330, 118, 122ecase23d 1472 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 206  wa 395  wo 847  w3o 1085  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  wss 3963  c0 4339  {ctp 4635  cop 4637   class class class wbr 5148   Or wor 5596  dom cdm 5689  cres 5691  Rel wrel 5694  Ord word 6385  Oncon0 6386  suc csuc 6388  Fun wfun 6557  cfv 6563  1oc1o 8498  2oc2o 8499   No csur 27699   <s cslt 27700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703
This theorem is referenced by:  noinfbnd1lem4  27786
  Copyright terms: Public domain W3C validator