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

Theorem nogt01o 27044
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 27024 . . . 4 <s Or No
2 simp11 1203 . . . 4 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐴 No )
3 sonr 5568 . . . 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 1227 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 <s 𝐵)
6 simpl2l 1226 . . . . 5 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐴𝑋) = (𝐵𝑋))
7 simpl11 1248 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 No )
8 nofun 26997 . . . . . . . 8 (𝐴 No → Fun 𝐴)
9 funrel 6518 . . . . . . . 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 1250 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝑋 ∈ On)
13 simpr 485 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐴𝑋) = ∅)
14 nolt02olem 27042 . . . . . . 7 ((𝐴 No 𝑋 ∈ On ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
157, 12, 13, 14syl3anc 1371 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → dom 𝐴𝑋)
16 relssres 5978 . . . . . 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 1249 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐵 No )
19 nofun 26997 . . . . . . . 8 (𝐵 No → Fun 𝐵)
20 funrel 6518 . . . . . . . 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 1193 . . . . . . 7 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → (𝐵𝑋) = ∅)
24 nolt02olem 27042 . . . . . . 7 ((𝐵 No 𝑋 ∈ On ∧ (𝐵𝑋) = ∅) → dom 𝐵𝑋)
2518, 12, 23, 24syl3anc 1371 . . . . . 6 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → dom 𝐵𝑋)
26 relssres 5978 . . . . . 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 2784 . . . 4 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 = 𝐵)
295, 28breqtrrd 5133 . . 3 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = ∅) → 𝐴 <s 𝐴)
304, 29mtand 814 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ¬ (𝐴𝑋) = ∅)
31 simp2r 1200 . . . . 5 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐴 <s 𝐵)
32 simp12 1204 . . . . . 6 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → 𝐵 No )
33 sltval 26995 . . . . . 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 231 . . . 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 3104 . . . . 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 217 . . 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 8434 . . . . . . . . . . . 12 1o ≠ ∅
4039neii 2945 . . . . . . . . . . 11 ¬ 1o = ∅
41 eqtr2 2760 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) → 1o = ∅)
4240, 41mto 196 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅)
43 df-2o 8413 . . . . . . . . . . . . 13 2o = suc 1o
44 2on 8426 . . . . . . . . . . . . . . . 16 2o ∈ On
4543, 44eqeltrri 2835 . . . . . . . . . . . . . . 15 suc 1o ∈ On
4645onordi 6428 . . . . . . . . . . . . . 14 Ord suc 1o
47 1oex 8422 . . . . . . . . . . . . . . 15 1o ∈ V
4847sucid 6399 . . . . . . . . . . . . . 14 1o ∈ suc 1o
49 nordeq 6336 . . . . . . . . . . . . . 14 ((Ord suc 1o ∧ 1o ∈ suc 1o) → suc 1o ≠ 1o)
5046, 48, 49mp2an 690 . . . . . . . . . . . . 13 suc 1o ≠ 1o
5143, 50eqnetri 3014 . . . . . . . . . . . 12 2o ≠ 1o
5251nesymi 3001 . . . . . . . . . . 11 ¬ 1o = 2o
53 eqtr2 2760 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) → 1o = 2o)
5452, 53mto 196 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o)
55 2on0 8428 . . . . . . . . . . . 12 2o ≠ ∅
5655nesymi 3001 . . . . . . . . . . 11 ¬ ∅ = 2o
57 eqtr2 2760 . . . . . . . . . . 11 ((((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o) → ∅ = 2o)
5856, 57mto 196 . . . . . . . . . 10 ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o)
5942, 54, 583pm3.2i 1339 . . . . . . . . 9 (¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴𝑋)‘𝑥) = 1o ∧ ((𝐴𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴𝑋)‘𝑥) = ∅ ∧ ((𝐴𝑋)‘𝑥) = 2o))
60 fvex 6855 . . . . . . . . . . . 12 ((𝐴𝑋)‘𝑥) ∈ V
6160, 60brtp 5480 . . . . . . . . . . 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 1109 . . . . . . . . . . 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 274 . . . . . . . . . 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 229 . . . . . . . 8 ¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴𝑋)‘𝑥)
66 simpl2l 1226 . . . . . . . . . . 11 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) → (𝐴𝑋) = (𝐵𝑋))
6766adantr 481 . . . . . . . . . 10 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = (𝐵𝑋))
6867fveq1d 6844 . . . . . . . . 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 5117 . . . . . . . 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 325 . . . . . . 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 6861 . . . . . . . . 9 (𝑥𝑋 → ((𝐴𝑋)‘𝑥) = (𝐴𝑥))
72 fvres 6861 . . . . . . . . 9 (𝑥𝑋 → ((𝐵𝑋)‘𝑥) = (𝐵𝑥))
7371, 72breq12d 5118 . . . . . . . 8 (𝑥𝑋 → (((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
7473notbid 317 . . . . . . 7 (𝑥𝑋 → (¬ ((𝐴𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵𝑋)‘𝑥) ↔ ¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
7570, 74syl5ibcom 244 . . . . . 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 2945 . . . . . . . . . . 11 ¬ 2o = 1o
7776intnanr 488 . . . . . . . . . 10 ¬ (2o = 1o ∧ ∅ = ∅)
7856intnan 487 . . . . . . . . . 10 ¬ (2o = 1o ∧ ∅ = 2o)
7956intnan 487 . . . . . . . . . 10 ¬ (2o = ∅ ∧ ∅ = 2o)
8077, 78, 793pm3.2i 1339 . . . . . . . . 9 (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o))
81 2oex 8423 . . . . . . . . . . . 12 2o ∈ V
82 0ex 5264 . . . . . . . . . . . 12 ∅ ∈ V
8381, 82brtp 5480 . . . . . . . . . . 11 (2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅ ↔ ((2o = 1o ∧ ∅ = ∅) ∨ (2o = 1o ∧ ∅ = 2o) ∨ (2o = ∅ ∧ ∅ = 2o)))
84 3oran 1109 . . . . . . . . . . 11 (((2o = 1o ∧ ∅ = ∅) ∨ (2o = 1o ∧ ∅ = 2o) ∨ (2o = ∅ ∧ ∅ = 2o)) ↔ ¬ (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)))
8583, 84bitri 274 . . . . . . . . . 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 229 . . . . . . . 8 ¬ 2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅
88 simplr 767 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐴𝑋) = 2o)
89 simpll3 1214 . . . . . . . . 9 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝐵𝑋) = ∅)
9088, 89breq12d 5118 . . . . . . . 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 326 . . . . . . 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 6842 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐴𝑥) = (𝐴𝑋))
93 fveq2 6842 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐵𝑥) = (𝐵𝑋))
9492, 93breq12d 5118 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
9594notbid 317 . . . . . . 7 (𝑥 = 𝑋 → (¬ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥) ↔ ¬ (𝐴𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑋)))
9691, 95syl5ibrcom 246 . . . . . 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 6842 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝐴𝑦) = (𝐴𝑋))
98 fveq2 6842 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝐵𝑦) = (𝐵𝑋))
9997, 98eqeq12d 2752 . . . . . . . . . . . 12 (𝑦 = 𝑋 → ((𝐴𝑦) = (𝐵𝑦) ↔ (𝐴𝑋) = (𝐵𝑋)))
10099rspccv 3578 . . . . . . . . . . 11 (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) → (𝑋𝑥 → (𝐴𝑋) = (𝐵𝑋)))
101100ad2antll 727 . . . . . . . . . 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 2743 . . . . . . . . . 10 ((𝐴𝑋) = (𝐵𝑋) ↔ (𝐵𝑋) = (𝐴𝑋))
103101, 102syl6ib 250 . . . . . . . . 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 2752 . . . . . . . . 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 238 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → (𝑋𝑥 → ∅ = 2o))
10656, 105mtoi 198 . . . . . . 7 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → ¬ 𝑋𝑥)
107 simprl 769 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑥 ∈ On)
108 simpl13 1250 . . . . . . . . 9 ((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) → 𝑋 ∈ On)
109108adantr 481 . . . . . . . 8 (((((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) ∧ (𝐴𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦))) → 𝑋 ∈ On)
110 ontri1 6351 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ ¬ 𝑋𝑥))
111 onsseleq 6358 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥𝑋 ↔ (𝑥𝑋𝑥 = 𝑋)))
112110, 111bitr3d 280 . . . . . . . 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 231 . . . . . 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 858 . . . . 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 457 . . . 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 3143 . . 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 814 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ¬ (𝐴𝑋) = 2o)
119 nofv 27005 . . . 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 1093 . . 3 (((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o) ↔ ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 2o))
122120, 121sylib 217 . 2 (((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵𝑋) = ∅) → ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 2o))
12330, 118, 122ecase23d 1473 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 205  wa 396  wo 845  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  wss 3910  c0 4282  {ctp 4590  cop 4592   class class class wbr 5105   Or wor 5544  dom cdm 5633  cres 5635  Rel wrel 5638  Ord word 6316  Oncon0 6317  suc csuc 6319  Fun wfun 6490  cfv 6496  1oc1o 8405  2oc2o 8406   No csur 26988   <s cslt 26989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992
This theorem is referenced by:  noinfbnd1lem4  27074
  Copyright terms: Public domain W3C validator