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Theorem sltsolem1 31951
Description: Lemma for sltso 31952. 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 7620 . . . . . . . 8 1𝑜 ≠ ∅
21neii 2825 . . . . . . 7 ¬ 1𝑜 = ∅
3 eqtr2 2671 . . . . . . 7 ((𝑥 = 1𝑜𝑥 = ∅) → 1𝑜 = ∅)
42, 3mto 188 . . . . . 6 ¬ (𝑥 = 1𝑜𝑥 = ∅)
5 1on 7612 . . . . . . . . 9 1𝑜 ∈ On
6 0elon 5816 . . . . . . . . 9 ∅ ∈ On
7 df-2o 7606 . . . . . . . . . . 11 2𝑜 = suc 1𝑜
8 df-1o 7605 . . . . . . . . . . 11 1𝑜 = suc ∅
97, 8eqeq12i 2665 . . . . . . . . . 10 (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
10 suc11 5869 . . . . . . . . . 10 ((1𝑜 ∈ On ∧ ∅ ∈ On) → (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅))
119, 10syl5bb 272 . . . . . . . . 9 ((1𝑜 ∈ On ∧ ∅ ∈ On) → (2𝑜 = 1𝑜 ↔ 1𝑜 = ∅))
125, 6, 11mp2an 708 . . . . . . . 8 (2𝑜 = 1𝑜 ↔ 1𝑜 = ∅)
131, 12nemtbir 2918 . . . . . . 7 ¬ 2𝑜 = 1𝑜
14 eqtr2 2671 . . . . . . . 8 ((𝑥 = 2𝑜𝑥 = 1𝑜) → 2𝑜 = 1𝑜)
1514ancoms 468 . . . . . . 7 ((𝑥 = 1𝑜𝑥 = 2𝑜) → 2𝑜 = 1𝑜)
1613, 15mto 188 . . . . . 6 ¬ (𝑥 = 1𝑜𝑥 = 2𝑜)
17 nsuceq0 5843 . . . . . . . 8 suc 1𝑜 ≠ ∅
187eqeq1i 2656 . . . . . . . 8 (2𝑜 = ∅ ↔ suc 1𝑜 = ∅)
1917, 18nemtbir 2918 . . . . . . 7 ¬ 2𝑜 = ∅
20 eqtr2 2671 . . . . . . . 8 ((𝑥 = 2𝑜𝑥 = ∅) → 2𝑜 = ∅)
2120ancoms 468 . . . . . . 7 ((𝑥 = ∅ ∧ 𝑥 = 2𝑜) → 2𝑜 = ∅)
2219, 21mto 188 . . . . . 6 ¬ (𝑥 = ∅ ∧ 𝑥 = 2𝑜)
234, 16, 223pm3.2ni 31720 . . . . 5 ¬ ((𝑥 = 1𝑜𝑥 = ∅) ∨ (𝑥 = 1𝑜𝑥 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑥 = 2𝑜))
24 vex 3234 . . . . . 6 𝑥 ∈ V
2524, 24brtp 31765 . . . . 5 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥 ↔ ((𝑥 = 1𝑜𝑥 = ∅) ∨ (𝑥 = 1𝑜𝑥 = 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 3234 . . . . . . 7 𝑦 ∈ V
2924, 28brtp 31765 . . . . . 6 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦 ↔ ((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
30 vex 3234 . . . . . . 7 𝑧 ∈ V
3128, 30brtp 31765 . . . . . 6 (𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧 ↔ ((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)))
32 eqtr2 2671 . . . . . . . . . . . . 13 ((𝑦 = 1𝑜𝑦 = ∅) → 1𝑜 = ∅)
332, 32mto 188 . . . . . . . . . . . 12 ¬ (𝑦 = 1𝑜𝑦 = ∅)
3433pm2.21i 116 . . . . . . . . . . 11 ((𝑦 = 1𝑜𝑦 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3534ad2ant2rl 800 . . . . . . . . . 10 (((𝑦 = 1𝑜𝑧 = ∅) ∧ (𝑥 = 1𝑜𝑦 = ∅)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3635expcom 450 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = ∅) → ((𝑦 = 1𝑜𝑧 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
3734ad2ant2rl 800 . . . . . . . . . 10 (((𝑦 = 1𝑜𝑧 = 2𝑜) ∧ (𝑥 = 1𝑜𝑦 = ∅)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
3837expcom 450 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = ∅) → ((𝑦 = 1𝑜𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
39 3mix2 1251 . . . . . . . . . . 11 ((𝑥 = 1𝑜𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4039ad2ant2rl 800 . . . . . . . . . 10 (((𝑥 = 1𝑜𝑦 = ∅) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4140ex 449 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = ∅) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
4236, 38, 413jaod 1432 . . . . . . . 8 ((𝑥 = 1𝑜𝑦 = ∅) → (((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
43 eqtr2 2671 . . . . . . . . . . . . 13 ((𝑦 = 2𝑜𝑦 = 1𝑜) → 2𝑜 = 1𝑜)
4413, 43mto 188 . . . . . . . . . . . 12 ¬ (𝑦 = 2𝑜𝑦 = 1𝑜)
4544pm2.21i 116 . . . . . . . . . . 11 ((𝑦 = 2𝑜𝑦 = 1𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4645ad2ant2lr 799 . . . . . . . . . 10 (((𝑥 = 1𝑜𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜𝑧 = ∅)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4746ex 449 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 2𝑜) → ((𝑦 = 1𝑜𝑧 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
4845ad2ant2lr 799 . . . . . . . . . 10 (((𝑥 = 1𝑜𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
4948ex 449 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 2𝑜) → ((𝑦 = 1𝑜𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
50 eqtr2 2671 . . . . . . . . . . . . 13 ((𝑦 = 2𝑜𝑦 = ∅) → 2𝑜 = ∅)
5119, 50mto 188 . . . . . . . . . . . 12 ¬ (𝑦 = 2𝑜𝑦 = ∅)
5251pm2.21i 116 . . . . . . . . . . 11 ((𝑦 = 2𝑜𝑦 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5352ad2ant2lr 799 . . . . . . . . . 10 (((𝑥 = 1𝑜𝑦 = 2𝑜) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5453ex 449 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 2𝑜) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5547, 49, 543jaod 1432 . . . . . . . 8 ((𝑥 = 1𝑜𝑦 = 2𝑜) → (((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5645ad2ant2lr 799 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜𝑧 = ∅)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5756ex 449 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = 1𝑜𝑧 = ∅) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
5845ad2ant2lr 799 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = 1𝑜𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
5958ex 449 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = 1𝑜𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6052ad2ant2lr 799 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2𝑜) ∧ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6160ex 449 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑦 = ∅ ∧ 𝑧 = 2𝑜) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6257, 59, 613jaod 1432 . . . . . . . 8 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → (((𝑦 = 1𝑜𝑧 = ∅) ∨ (𝑦 = 1𝑜𝑧 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑧 = 2𝑜)) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜))))
6342, 55, 623jaoi 1431 . . . . . . 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 444 . . . . . 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 495 . . . . 5 ((𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧) → ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6624, 30brtp 31765 . . . . 5 (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑧 ↔ ((𝑥 = 1𝑜𝑧 = ∅) ∨ (𝑥 = 1𝑜𝑧 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑧 = 2𝑜)))
6765, 66sylibr 224 . . . 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 4262 . . . . 5 (𝑥 ∈ {1𝑜, 2𝑜, ∅} ↔ (𝑥 = 1𝑜𝑥 = 2𝑜𝑥 = ∅))
7028eltp 4262 . . . . 5 (𝑦 ∈ {1𝑜, 2𝑜, ∅} ↔ (𝑦 = 1𝑜𝑦 = 2𝑜𝑦 = ∅))
71 eqtr3 2672 . . . . . . . . . 10 ((𝑥 = 1𝑜𝑦 = 1𝑜) → 𝑥 = 𝑦)
72713mix2d 1257 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 1𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7372ex 449 . . . . . . . 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 1251 . . . . . . . . . 10 ((𝑥 = 1𝑜𝑦 = 2𝑜) → ((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
75743mix1d 1256 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7675ex 449 . . . . . . . 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 1250 . . . . . . . . . 10 ((𝑥 = 1𝑜𝑦 = ∅) → ((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
78773mix1d 1256 . . . . . . . . 9 ((𝑥 = 1𝑜𝑦 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
7978ex 449 . . . . . . . 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 1432 . . . . . . 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 1251 . . . . . . . . . 10 ((𝑦 = 1𝑜𝑥 = 2𝑜) → ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
82813mix3d 1258 . . . . . . . . 9 ((𝑦 = 1𝑜𝑥 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8382expcom 450 . . . . . . . 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 2672 . . . . . . . . . 10 ((𝑥 = 2𝑜𝑦 = 2𝑜) → 𝑥 = 𝑦)
85843mix2d 1257 . . . . . . . . 9 ((𝑥 = 2𝑜𝑦 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8685ex 449 . . . . . . . 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 1252 . . . . . . . . . 10 ((𝑦 = ∅ ∧ 𝑥 = 2𝑜) → ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
88873mix3d 1258 . . . . . . . . 9 ((𝑦 = ∅ ∧ 𝑥 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
8988expcom 450 . . . . . . . 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 1432 . . . . . . 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 1250 . . . . . . . . . 10 ((𝑦 = 1𝑜𝑥 = ∅) → ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
92913mix3d 1258 . . . . . . . . 9 ((𝑦 = 1𝑜𝑥 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9392expcom 450 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = 1𝑜 → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
94 3mix3 1252 . . . . . . . . . 10 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → ((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)))
95943mix1d 1256 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2𝑜) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9695ex 449 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = 2𝑜 → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
97 eqtr3 2672 . . . . . . . . . 10 ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑥 = 𝑦)
98973mix2d 1257 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜))))
9998ex 449 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = ∅ → (((𝑥 = 1𝑜𝑦 = ∅) ∨ (𝑥 = 1𝑜𝑦 = 2𝑜) ∨ (𝑥 = ∅ ∧ 𝑦 = 2𝑜)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))))
10093, 96, 993jaod 1432 . . . . . . 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 1431 . . . . . 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 444 . . . . 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 495 . . . 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 251 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
10528, 24brtp 31765 . . . . 5 (𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥 ↔ ((𝑦 = 1𝑜𝑥 = ∅) ∨ (𝑦 = 1𝑜𝑥 = 2𝑜) ∨ (𝑦 = ∅ ∧ 𝑥 = 2𝑜)))
10629, 104, 1053orbi123i 1271 . . . 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 224 . . 3 ((𝑥 ∈ {1𝑜, 2𝑜, ∅} ∧ 𝑦 ∈ {1𝑜, 2𝑜, ∅}) → (𝑥{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑦𝑥 = 𝑦𝑦{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩}𝑥))
10827, 68, 107issoi 5095 . 2 {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or {1𝑜, 2𝑜, ∅}
109 df-tp 4215 . . 3 {1𝑜, 2𝑜, ∅} = ({1𝑜, 2𝑜} ∪ {∅})
110 soeq2 5084 . . 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 220 1 {⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  cun 3605  c0 3948  {csn 4210  {cpr 4212  {ctp 4214  cop 4216   class class class wbr 4685   Or wor 5063  Oncon0 5761  suc csuc 5763  1𝑜c1o 7598  2𝑜c2o 7599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767  df-1o 7605  df-2o 7606
This theorem is referenced by:  sltso  31952
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