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Theorem sltsolem1 31067
Description: Lemma for sltso 31068. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)
Assertion
Ref Expression
sltsolem1 {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅})
Proof of Theorem sltsolem1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 7462 . . . . . . . 8 1𝑜 ≠ ∅
21neii 2784 . . . . . . 7 ¬ 1𝑜 = ∅
3 eqtr2 2630 . . . . . . 7 ((𝑥 = 1𝑜 ∧ 𝑥 = ∅) → 1𝑜 = ∅)
42, 3mto 187 . . . . . 6 ¬ (𝑥 = 1𝑜 ∧ 𝑥 = ∅)
5 1on 7454 . . . . . . . . 9 1𝑜 ∈ On
6 0elon 5695 . . . . . . . . 9 ∅ ∈ On
7 df-2o 7448 . . . . . . . . . . 11 2𝑜 = suc 1𝑜
8 df-1o 7447 . . . . . . . . . . 11 1𝑜 = suc ∅
97, 8eqeq12i 2624 . . . . . . . . . 10 (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
10 suc11 5748 . . . . . . . . . 10 ((1𝑜 ∈ On ∧ ∅ ∈ On) → (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅))
119, 10syl5bb 271 . . . . . . . . 9 ((1𝑜 ∈ On ∧ ∅ ∈ On) → (2𝑜 = 1𝑜 ↔ 1𝑜 = ∅))
125, 6, 11mp2an 704 . . . . . . . 8 (2𝑜 = 1𝑜 ↔ 1𝑜 = ∅)
131, 12nemtbir 2877 . . . . . . 7 ¬ 2𝑜 = 1𝑜
14 eqtr2 2630 . . . . . . . 8 ((𝑥 = 2𝑜 ∧ 𝑥 = 1𝑜) → 2𝑜 = 1𝑜)
1514ancoms 468 . . . . . . 7 ((𝑥 = 1𝑜 ∧ 𝑥 = 2𝑜) → 2𝑜 = 1𝑜)
1613, 15mto 187 . . . . . 6 ¬ (𝑥 = 1𝑜 ∧ 𝑥 = 2𝑜)
17 nsuceq0 5722 . . . . . . . 8 suc 1𝑜 ≠ ∅
187eqeq1i 2615 . . . . . . . 8 (2𝑜 = ∅ ↔ suc 1𝑜 = ∅)
1917, 18nemtbir 2877 . . . . . . 7 ¬ 2𝑜 = ∅
20 eqtr2 2630 . . . . . . . 8 ((𝑥 = 2𝑜 ∧ 𝑥 = ∅) → 2𝑜 = ∅)
2120ancoms 468 . . . . . . 7 ((𝑥 = ∅ ∧ 𝑥 = 2𝑜) → 2𝑜 = ∅)
2219, 21mto 187 . . . . . 6 ¬ (𝑥 = ∅ ∧ 𝑥 = 2𝑜)
234, 16, 223pm3.2ni 30849 . . . . 5 ¬ ((𝑥 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑥 = 2𝑜))
24 vex 3176 . . . . . 6 𝑥 ∈ V
2524, 24brtp 30892 . . . . 5 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥 ↔ ((𝑥 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑥 = 2𝑜)))
2623, 25mtbir 312 . . . 4 ¬ 𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥
2726a1i 11 . . 3 (𝑥 ∈ {1𝑜, 2𝑜, ∅} → ¬ 𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥)
28 vex 3176 . . . . . . 7 𝑦 ∈ V
2924, 28brtp 30892 . . . . . 6 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦 ↔ ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
30 vex 3176 . . . . . . 7 𝑧 ∈ V
3128, 30brtp 30892 . . . . . 6 (𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧 ↔ ((𝑦 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)))
32 eqtr2 2630 . . . . . . . . . . . . 13 ((𝑦 = 1𝑜 ∧ 𝑦 = ∅) → 1𝑜 = ∅)
332, 32mto 187 . . . . . . . . . . . 12 ¬ (𝑦 = 1𝑜 ∧ 𝑦 = ∅)
3433pm2.21i 115 . . . . . . . . . . 11 ((𝑦 = 1𝑜 ∧ 𝑦 = ∅) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3534ad2ant2rl 781 . . . . . . . . . 10 (((𝑦 = 1𝑜 ∧ 𝑧 = ∅) ∧ (𝑥 = 1𝑜 ∧ 𝑦 = ∅)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3635expcom 450 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) → ((𝑦 = 1𝑜 ∧ 𝑧 = ∅) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
3734ad2ant2rl 781 . . . . . . . . . 10 (((𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) ∧ (𝑥 = 1𝑜 ∧ 𝑦 = ∅)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3837expcom 450 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) → ((𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
39 3mix2 1224 . . . . . . . . . . 11 ((𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4039ad2ant2rl 781 . . . . . . . . . 10 (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4140ex 449 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
4236, 38, 413jaod 1384 . . . . . . . 8 ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) → (((𝑦 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
43 eqtr2 2630 . . . . . . . . . . . . 13 ((𝑦 = 2𝑜 ∧ 𝑦 = 1𝑜) → 2𝑜 = 1𝑜)
4413, 43mto 187 . . . . . . . . . . . 12 ¬ (𝑦 = 2𝑜 ∧ 𝑦 = 1𝑜)
4544pm2.21i 115 . . . . . . . . . . 11 ((𝑦 = 2𝑜 ∧ 𝑦 = 1𝑜) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4645ad2ant2lr 780 . . . . . . . . . 10 (((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜 ∧ 𝑧 = ∅)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4746ex 449 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) → ((𝑦 = 1𝑜 ∧ 𝑧 = ∅) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
4845ad2ant2lr 780 . . . . . . . . . 10 (((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4948ex 449 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) → ((𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
50 eqtr2 2630 . . . . . . . . . . . . 13 ((𝑦 = 2𝑜 ∧ 𝑦 = ∅) → 2𝑜 = ∅)
5119, 50mto 187 . . . . . . . . . . . 12 ¬ (𝑦 = 2𝑜 ∧ 𝑦 = ∅)
5251pm2.21i 115 . . . . . . . . . . 11 ((𝑦 = 2𝑜 ∧ 𝑦 = ∅) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5352ad2ant2lr 780 . . . . . . . . . 10 (((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5453ex 449 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5547, 49, 543jaod 1384 . . . . . . . 8 ((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) → (((𝑦 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5645ad2ant2lr 780 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜 ∧ 𝑧 = ∅)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5756ex 449 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = 1𝑜 ∧ 𝑧 = ∅) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5845ad2ant2lr 780 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5958ex 449 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6052ad2ant2lr 780 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6160ex 449 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6257, 59, 613jaod 1384 . . . . . . . 8 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → (((𝑦 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6342, 55, 623jaoi 1383 . . . . . . 7 (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) → (((𝑦 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6463imp 444 . . . . . 6 ((((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∧ ((𝑦 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜))) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6529, 31, 64syl2anb 495 . . . . 5 ((𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦 ∧ 𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧) → ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6624, 30brtp 30892 . . . . 5 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧 ↔ ((𝑥 = 1𝑜 ∧ 𝑧 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6765, 66sylibr 223 . . . 4 ((𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦 ∧ 𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧) → 𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧)
6867a1i 11 . . 3 ((𝑥 ∈ {1𝑜, 2𝑜, ∅} ∧ 𝑦 ∈ {1𝑜, 2𝑜, ∅} ∧ 𝑧 ∈ {1𝑜, 2𝑜, ∅}) → ((𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦 ∧ 𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧) → 𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧))
6924eltp 4177 . . . . 5 (𝑥 ∈ {1𝑜, 2𝑜, ∅} ↔ (𝑥 = 1𝑜 ∨ 𝑥 = 2𝑜 ∨ 𝑥 = ∅))
7028eltp 4177 . . . . 5 (𝑦 ∈ {1𝑜, 2𝑜, ∅} ↔ (𝑦 = 1𝑜 ∨ 𝑦 = 2𝑜 ∨ 𝑦 = ∅))
71 eqtr3 2631 . . . . . . . . . 10 ((𝑥 = 1𝑜 ∧ 𝑦 = 1𝑜) → 𝑥 = 𝑦)
72713mix2d 1230 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = 1𝑜) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7372ex 449 . . . . . . . 8 (𝑥 = 1𝑜 → (𝑦 = 1𝑜 → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
74 3mix2 1224 . . . . . . . . . 10 ((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
75743mix1d 1229 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7675ex 449 . . . . . . . 8 (𝑥 = 1𝑜 → (𝑦 = 2𝑜 → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
77 3mix1 1223 . . . . . . . . . 10 ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) → ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
78773mix1d 1229 . . . . . . . . 9 ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7978ex 449 . . . . . . . 8 (𝑥 = 1𝑜 → (𝑦 = ∅ → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
8073, 76, 793jaod 1384 . . . . . . 7 (𝑥 = 1𝑜 → ((𝑦 = 1𝑜 ∨ 𝑦 = 2𝑜 ∨ 𝑦 = ∅) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
81 3mix2 1224 . . . . . . . . . 10 ((𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) → ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
82813mix3d 1231 . . . . . . . . 9 ((𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8382expcom 450 . . . . . . . 8 (𝑥 = 2𝑜 → (𝑦 = 1𝑜 → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
84 eqtr3 2631 . . . . . . . . . 10 ((𝑥 = 2𝑜 ∧ 𝑦 = 2𝑜) → 𝑥 = 𝑦)
85843mix2d 1230 . . . . . . . . 9 ((𝑥 = 2𝑜 ∧ 𝑦 = 2𝑜) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8685ex 449 . . . . . . . 8 (𝑥 = 2𝑜 → (𝑦 = 2𝑜 → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
87 3mix3 1225 . . . . . . . . . 10 ((𝑦 = ∅ ∧ 𝑥 = 2𝑜) → ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
88873mix3d 1231 . . . . . . . . 9 ((𝑦 = ∅ ∧ 𝑥 = 2𝑜) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8988expcom 450 . . . . . . . 8 (𝑥 = 2𝑜 → (𝑦 = ∅ → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
9083, 86, 893jaod 1384 . . . . . . 7 (𝑥 = 2𝑜 → ((𝑦 = 1𝑜 ∨ 𝑦 = 2𝑜 ∨ 𝑦 = ∅) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
91 3mix1 1223 . . . . . . . . . 10 ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) → ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
92913mix3d 1231 . . . . . . . . 9 ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9392expcom 450 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = 1𝑜 → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
94 3mix3 1225 . . . . . . . . . 10 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
95943mix1d 1229 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9695ex 449 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = 2𝑜 → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
97 eqtr3 2631 . . . . . . . . . 10 ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑥 = 𝑦)
98973mix2d 1230 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9998ex 449 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = ∅ → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
10093, 96, 993jaod 1384 . . . . . . 7 (𝑥 = ∅ → ((𝑦 = 1𝑜 ∨ 𝑦 = 2𝑜 ∨ 𝑦 = ∅) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
10180, 90, 1003jaoi 1383 . . . . . 6 ((𝑥 = 1𝑜 ∨ 𝑥 = 2𝑜 ∨ 𝑥 = ∅) → ((𝑦 = 1𝑜 ∨ 𝑦 = 2𝑜 ∨ 𝑦 = ∅) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
102101imp 444 . . . . 5 (((𝑥 = 1𝑜 ∨ 𝑥 = 2𝑜 ∨ 𝑥 = ∅) ∧ (𝑦 = 1𝑜 ∨ 𝑦 = 2𝑜 ∨ 𝑦 = ∅)) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
10369, 70, 102syl2anb 495 . . . 4 ((𝑥 ∈ {1𝑜, 2𝑜, ∅} ∧ 𝑦 ∈ {1𝑜, 2𝑜, ∅}) → (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
104 biid 250 . . . . 5 (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
10528, 24brtp 30892 . . . . 5 (𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥 ↔ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
10629, 104, 1053orbi123i 1245 . . . 4 ((𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥) ↔ (((𝑥 = 1𝑜 ∧ 𝑦 = ∅) ∨ (𝑥 = 1𝑜 ∧ 𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜 ∧ 𝑥 = ∅) ∨ (𝑦 = 1𝑜 ∧ 𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
107103, 106sylibr 223 . . 3 ((𝑥 ∈ {1𝑜, 2𝑜, ∅} ∧ 𝑦 ∈ {1𝑜, 2𝑜, ∅}) → (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥))
10827, 68, 107issoi 4990 . 2 {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or {1𝑜, 2𝑜, ∅}
109 df-tp 4130 . . 3 {1𝑜, 2𝑜, ∅} = ({1𝑜, 2𝑜} ∪ {∅})
110 soeq2 4979 . . 3 ({1𝑜, 2𝑜, ∅} = ({1𝑜, 2𝑜} ∪ {∅}) → ({⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or {1𝑜, 2𝑜, ∅} ↔ {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅})))
111109, 110ax-mp 5 . 2 ({⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or {1𝑜, 2𝑜, ∅} ↔ {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅}))
112108, 111mpbi 219 1 {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  ∅c0 3874  {csn 4125  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583   Or wor 4958  Oncon0 5640  suc csuc 5642  1𝑜c1o 7440  2𝑜c2o 7441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646  df-1o 7447  df-2o 7448
This theorem is referenced by:  sltso  31068
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