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Theorem sltsolem1 30903
Description: Lemma for sltso 30904. 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 7337 . . . . . . . 8 1𝑜 ≠ ∅
21neii 2688 . . . . . . 7 ¬ 1𝑜 = ∅
3 eqtr2 2534 . . . . . . 7 ((𝑥 = 1𝑜𝑥 = ∅) → 1𝑜 = ∅)
42, 3mto 186 . . . . . 6 ¬ (𝑥 = 1𝑜𝑥 = ∅)
5 1on 7329 . . . . . . . . 9 1𝑜 ∈ On
6 0elon 5583 . . . . . . . . 9 ∅ ∈ On
7 df-2o 7323 . . . . . . . . . . 11 2𝑜 = suc 1𝑜
8 df-1o 7322 . . . . . . . . . . 11 1𝑜 = suc ∅
97, 8eqeq12i 2528 . . . . . . . . . 10 (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
10 suc11 5633 . . . . . . . . . 10 ((1𝑜 ∈ On ∧ ∅ ∈ On) → (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅))
119, 10syl5bb 270 . . . . . . . . 9 ((1𝑜 ∈ On ∧ ∅ ∈ On) → (2𝑜 = 1𝑜 ↔ 1𝑜 = ∅))
125, 6, 11mp2an 703 . . . . . . . 8 (2𝑜 = 1𝑜 ↔ 1𝑜 = ∅)
131, 12nemtbir 2781 . . . . . . 7 ¬ 2𝑜 = 1𝑜
14 eqtr2 2534 . . . . . . . 8 ((𝑥 = 2𝑜𝑥 = 1𝑜) → 2𝑜 = 1𝑜)
1514ancoms 467 . . . . . . 7 ((𝑥 = 1𝑜𝑥 = 2𝑜) → 2𝑜 = 1𝑜)
1613, 15mto 186 . . . . . 6 ¬ (𝑥 = 1𝑜𝑥 = 2𝑜)
17 nsuceq0 5610 . . . . . . . 8 suc 1𝑜 ≠ ∅
187eqeq1i 2519 . . . . . . . 8 (2𝑜 = ∅ ↔ suc 1𝑜 = ∅)
1917, 18nemtbir 2781 . . . . . . 7 ¬ 2𝑜 = ∅
20 eqtr2 2534 . . . . . . . 8 ((𝑥 = 2𝑜𝑥 = ∅) → 2𝑜 = ∅)
2120ancoms 467 . . . . . . 7 ((𝑥 = ∅ ∧ 𝑥 = 2𝑜) → 2𝑜 = ∅)
2219, 21mto 186 . . . . . 6 ¬ (𝑥 = ∅ ∧ 𝑥 = 2𝑜)
234, 16, 223pm3.2ni 30693 . . . . 5 ¬ ((𝑥 = 1𝑜𝑥 = ∅) ∨ (𝑥 = 1𝑜𝑥 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑥 = 2𝑜))
24 vex 3080 . . . . . 6 𝑥 ∈ V
2524, 24brtp 30735 . . . . 5 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥 ↔ ((𝑥 = 1𝑜𝑥 = ∅) ∨ (𝑥 = 1𝑜𝑥 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑥 = 2𝑜)))
2623, 25mtbir 311 . . . 4 ¬ 𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥
2726a1i 11 . . 3 (𝑥 ∈ {1𝑜, 2𝑜, ∅} → ¬ 𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥)
28 vex 3080 . . . . . . 7 𝑦 ∈ V
2924, 28brtp 30735 . . . . . 6 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦 ↔ ((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
30 vex 3080 . . . . . . 7 𝑧 ∈ V
3128, 30brtp 30735 . . . . . 6 (𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧 ↔ ((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)))
32 eqtr2 2534 . . . . . . . . . . . . 13 ((𝑦 = 1𝑜𝑦 = ∅) → 1𝑜 = ∅)
332, 32mto 186 . . . . . . . . . . . 12 ¬ (𝑦 = 1𝑜𝑦 = ∅)
3433pm2.21i 114 . . . . . . . . . . 11 ((𝑦 = 1𝑜𝑦 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3534ad2ant2rl 780 . . . . . . . . . 10 (((𝑦 = 1𝑜𝑧 = ∅) ∧ (𝑥 = 1𝑜𝑦 = ∅)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3635expcom 449 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = ∅) → ((𝑦 = 1𝑜𝑧 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
3734ad2ant2rl 780 . . . . . . . . . 10 (((𝑦 = 1𝑜𝑧 = 2𝑜) ∧ (𝑥 = 1𝑜𝑦 = ∅)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3837expcom 449 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = ∅) → ((𝑦 = 1𝑜𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
39 3mix2 1223 . . . . . . . . . . 11 ((𝑥 = 1𝑜𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4039ad2ant2rl 780 . . . . . . . . . 10 (((𝑥 = 1𝑜𝑦 = ∅) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4140ex 448 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = ∅) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
4236, 38, 413jaod 1383 . . . . . . . 8 ((𝑥 = 1𝑜𝑦 = ∅) → (((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
43 eqtr2 2534 . . . . . . . . . . . . 13 ((𝑦 = 2𝑜𝑦 = 1𝑜) → 2𝑜 = 1𝑜)
4413, 43mto 186 . . . . . . . . . . . 12 ¬ (𝑦 = 2𝑜𝑦 = 1𝑜)
4544pm2.21i 114 . . . . . . . . . . 11 ((𝑦 = 2𝑜𝑦 = 1𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4645ad2ant2lr 779 . . . . . . . . . 10 (((𝑥 = 1𝑜𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜𝑧 = ∅)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4746ex 448 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 2𝑜) → ((𝑦 = 1𝑜𝑧 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
4845ad2ant2lr 779 . . . . . . . . . 10 (((𝑥 = 1𝑜𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4948ex 448 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 2𝑜) → ((𝑦 = 1𝑜𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
50 eqtr2 2534 . . . . . . . . . . . . 13 ((𝑦 = 2𝑜𝑦 = ∅) → 2𝑜 = ∅)
5119, 50mto 186 . . . . . . . . . . . 12 ¬ (𝑦 = 2𝑜𝑦 = ∅)
5251pm2.21i 114 . . . . . . . . . . 11 ((𝑦 = 2𝑜𝑦 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5352ad2ant2lr 779 . . . . . . . . . 10 (((𝑥 = 1𝑜𝑦 = 2𝑜) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5453ex 448 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 2𝑜) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5547, 49, 543jaod 1383 . . . . . . . 8 ((𝑥 = 1𝑜𝑦 = 2𝑜) → (((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5645ad2ant2lr 779 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜𝑧 = ∅)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5756ex 448 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = 1𝑜𝑧 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5845ad2ant2lr 779 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5958ex 448 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = 1𝑜𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6052ad2ant2lr 779 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6160ex 448 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6257, 59, 613jaod 1383 . . . . . . . 8 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → (((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6342, 55, 623jaoi 1382 . . . . . . 7 (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) → (((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6463imp 443 . . . . . 6 ((((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∧ ((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜))) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6529, 31, 64syl2anb 494 . . . . 5 ((𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6624, 30brtp 30735 . . . . 5 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧 ↔ ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6765, 66sylibr 222 . . . 4 ((𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦𝑦{⟨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⟩}𝑦𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧) → 𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧))
6924eltp 4080 . . . . 5 (𝑥 ∈ {1𝑜, 2𝑜, ∅} ↔ (𝑥 = 1𝑜𝑥 = 2𝑜𝑥 = ∅))
7028eltp 4080 . . . . 5 (𝑦 ∈ {1𝑜, 2𝑜, ∅} ↔ (𝑦 = 1𝑜𝑦 = 2𝑜𝑦 = ∅))
71 eqtr3 2535 . . . . . . . . . 10 ((𝑥 = 1𝑜𝑦 = 1𝑜) → 𝑥 = 𝑦)
72713mix2d 1229 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 1𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7372ex 448 . . . . . . . 8 (𝑥 = 1𝑜 → (𝑦 = 1𝑜 → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
74 3mix2 1223 . . . . . . . . . 10 ((𝑥 = 1𝑜𝑦 = 2𝑜) → ((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
75743mix1d 1228 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7675ex 448 . . . . . . . 8 (𝑥 = 1𝑜 → (𝑦 = 2𝑜 → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
77 3mix1 1222 . . . . . . . . . 10 ((𝑥 = 1𝑜𝑦 = ∅) → ((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
78773mix1d 1228 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7978ex 448 . . . . . . . 8 (𝑥 = 1𝑜 → (𝑦 = ∅ → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
8073, 76, 793jaod 1383 . . . . . . 7 (𝑥 = 1𝑜 → ((𝑦 = 1𝑜𝑦 = 2𝑜𝑦 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
81 3mix2 1223 . . . . . . . . . 10 ((𝑦 = 1𝑜𝑥 = 2𝑜) → ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
82813mix3d 1230 . . . . . . . . 9 ((𝑦 = 1𝑜𝑥 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8382expcom 449 . . . . . . . 8 (𝑥 = 2𝑜 → (𝑦 = 1𝑜 → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
84 eqtr3 2535 . . . . . . . . . 10 ((𝑥 = 2𝑜𝑦 = 2𝑜) → 𝑥 = 𝑦)
85843mix2d 1229 . . . . . . . . 9 ((𝑥 = 2𝑜𝑦 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8685ex 448 . . . . . . . 8 (𝑥 = 2𝑜 → (𝑦 = 2𝑜 → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
87 3mix3 1224 . . . . . . . . . 10 ((𝑦 = ∅ ∧ 𝑥 = 2𝑜) → ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
88873mix3d 1230 . . . . . . . . 9 ((𝑦 = ∅ ∧ 𝑥 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8988expcom 449 . . . . . . . 8 (𝑥 = 2𝑜 → (𝑦 = ∅ → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
9083, 86, 893jaod 1383 . . . . . . 7 (𝑥 = 2𝑜 → ((𝑦 = 1𝑜𝑦 = 2𝑜𝑦 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
91 3mix1 1222 . . . . . . . . . 10 ((𝑦 = 1𝑜𝑥 = ∅) → ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
92913mix3d 1230 . . . . . . . . 9 ((𝑦 = 1𝑜𝑥 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9392expcom 449 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = 1𝑜 → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
94 3mix3 1224 . . . . . . . . . 10 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
95943mix1d 1228 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9695ex 448 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = 2𝑜 → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
97 eqtr3 2535 . . . . . . . . . 10 ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑥 = 𝑦)
98973mix2d 1229 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9998ex 448 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = ∅ → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
10093, 96, 993jaod 1383 . . . . . . 7 (𝑥 = ∅ → ((𝑦 = 1𝑜𝑦 = 2𝑜𝑦 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
10180, 90, 1003jaoi 1382 . . . . . 6 ((𝑥 = 1𝑜𝑥 = 2𝑜𝑥 = ∅) → ((𝑦 = 1𝑜𝑦 = 2𝑜𝑦 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
102101imp 443 . . . . 5 (((𝑥 = 1𝑜𝑥 = 2𝑜𝑥 = ∅) ∧ (𝑦 = 1𝑜𝑦 = 2𝑜𝑦 = ∅)) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
10369, 70, 102syl2anb 494 . . . 4 ((𝑥 ∈ {1𝑜, 2𝑜, ∅} ∧ 𝑦 ∈ {1𝑜, 2𝑜, ∅}) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
104 biid 249 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
10528, 24brtp 30735 . . . . 5 (𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥 ↔ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
10629, 104, 1053orbi123i 1244 . . . 4 ((𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦𝑥 = 𝑦𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥) ↔ (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
107103, 106sylibr 222 . . 3 ((𝑥 ∈ {1𝑜, 2𝑜, ∅} ∧ 𝑦 ∈ {1𝑜, 2𝑜, ∅}) → (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦𝑥 = 𝑦𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥))
10827, 68, 107issoi 4884 . 2 {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or {1𝑜, 2𝑜, ∅}
109 df-tp 4033 . . 3 {1𝑜, 2𝑜, ∅} = ({1𝑜, 2𝑜} ∪ {∅})
110 soeq2 4873 . . 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 218 1 {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3o 1029  w3a 1030   = wceq 1474  wcel 1938  cun 3442  c0 3777  {csn 4028  {cpr 4030  {ctp 4032  cop 4034   class class class wbr 4481   Or wor 4852  Oncon0 5530  suc csuc 5532  1𝑜c1o 7315  2𝑜c2o 7316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-tr 4579  df-eprel 4843  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-ord 5533  df-on 5534  df-suc 5536  df-1o 7322  df-2o 7323
This theorem is referenced by:  sltso  30904
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