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Theorem sltsolem1 33064
 Description: Lemma for sltso 33065. 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 8113 . . . . . . . 8 1o ≠ ∅
21neii 3022 . . . . . . 7 ¬ 1o = ∅
3 eqtr2 2846 . . . . . . 7 ((𝑥 = 1o𝑥 = ∅) → 1o = ∅)
42, 3mto 198 . . . . . 6 ¬ (𝑥 = 1o𝑥 = ∅)
5 1on 8103 . . . . . . . . 9 1o ∈ On
6 0elon 6241 . . . . . . . . 9 ∅ ∈ On
7 df-2o 8097 . . . . . . . . . . 11 2o = suc 1o
8 df-1o 8096 . . . . . . . . . . 11 1o = suc ∅
97, 8eqeq12i 2840 . . . . . . . . . 10 (2o = 1o ↔ suc 1o = suc ∅)
10 suc11 6291 . . . . . . . . . 10 ((1o ∈ On ∧ ∅ ∈ On) → (suc 1o = suc ∅ ↔ 1o = ∅))
119, 10syl5bb 284 . . . . . . . . 9 ((1o ∈ On ∧ ∅ ∈ On) → (2o = 1o ↔ 1o = ∅))
125, 6, 11mp2an 688 . . . . . . . 8 (2o = 1o ↔ 1o = ∅)
131, 12nemtbir 3116 . . . . . . 7 ¬ 2o = 1o
14 eqtr2 2846 . . . . . . . 8 ((𝑥 = 2o𝑥 = 1o) → 2o = 1o)
1514ancoms 459 . . . . . . 7 ((𝑥 = 1o𝑥 = 2o) → 2o = 1o)
1613, 15mto 198 . . . . . 6 ¬ (𝑥 = 1o𝑥 = 2o)
17 nsuceq0 6268 . . . . . . . 8 suc 1o ≠ ∅
187eqeq1i 2830 . . . . . . . 8 (2o = ∅ ↔ suc 1o = ∅)
1917, 18nemtbir 3116 . . . . . . 7 ¬ 2o = ∅
20 eqtr2 2846 . . . . . . . 8 ((𝑥 = 2o𝑥 = ∅) → 2o = ∅)
2120ancoms 459 . . . . . . 7 ((𝑥 = ∅ ∧ 𝑥 = 2o) → 2o = ∅)
2219, 21mto 198 . . . . . 6 ¬ (𝑥 = ∅ ∧ 𝑥 = 2o)
234, 16, 223pm3.2ni 32827 . . . . 5 ¬ ((𝑥 = 1o𝑥 = ∅) ∨ (𝑥 = 1o𝑥 = 2o) ∨ (𝑥 = ∅ ∧ 𝑥 = 2o))
24 vex 3502 . . . . . 6 𝑥 ∈ V
2524, 24brtp 32869 . . . . 5 (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥 ↔ ((𝑥 = 1o𝑥 = ∅) ∨ (𝑥 = 1o𝑥 = 2o) ∨ (𝑥 = ∅ ∧ 𝑥 = 2o)))
2623, 25mtbir 324 . . . 4 ¬ 𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥
2726a1i 11 . . 3 (𝑥 ∈ {1o, 2o, ∅} → ¬ 𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥)
28 vex 3502 . . . . . . 7 𝑦 ∈ V
2924, 28brtp 32869 . . . . . 6 (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑦 ↔ ((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
30 vex 3502 . . . . . . 7 𝑧 ∈ V
3128, 30brtp 32869 . . . . . 6 (𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧 ↔ ((𝑦 = 1o𝑧 = ∅) ∨ (𝑦 = 1o𝑧 = 2o) ∨ (𝑦 = ∅ ∧ 𝑧 = 2o)))
32 eqtr2 2846 . . . . . . . . . . . . 13 ((𝑦 = 1o𝑦 = ∅) → 1o = ∅)
332, 32mto 198 . . . . . . . . . . . 12 ¬ (𝑦 = 1o𝑦 = ∅)
3433pm2.21i 119 . . . . . . . . . . 11 ((𝑦 = 1o𝑦 = ∅) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
3534ad2ant2rl 745 . . . . . . . . . 10 (((𝑦 = 1o𝑧 = ∅) ∧ (𝑥 = 1o𝑦 = ∅)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
3635expcom 414 . . . . . . . . 9 ((𝑥 = 1o𝑦 = ∅) → ((𝑦 = 1o𝑧 = ∅) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
3734ad2ant2rl 745 . . . . . . . . . 10 (((𝑦 = 1o𝑧 = 2o) ∧ (𝑥 = 1o𝑦 = ∅)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
3837expcom 414 . . . . . . . . 9 ((𝑥 = 1o𝑦 = ∅) → ((𝑦 = 1o𝑧 = 2o) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
39 3mix2 1325 . . . . . . . . . . 11 ((𝑥 = 1o𝑧 = 2o) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
4039ad2ant2rl 745 . . . . . . . . . 10 (((𝑥 = 1o𝑦 = ∅) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
4140ex 413 . . . . . . . . 9 ((𝑥 = 1o𝑦 = ∅) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
4236, 38, 413jaod 1422 . . . . . . . 8 ((𝑥 = 1o𝑦 = ∅) → (((𝑦 = 1o𝑧 = ∅) ∨ (𝑦 = 1o𝑧 = 2o) ∨ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
43 eqtr2 2846 . . . . . . . . . . . . 13 ((𝑦 = 2o𝑦 = 1o) → 2o = 1o)
4413, 43mto 198 . . . . . . . . . . . 12 ¬ (𝑦 = 2o𝑦 = 1o)
4544pm2.21i 119 . . . . . . . . . . 11 ((𝑦 = 2o𝑦 = 1o) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
4645ad2ant2lr 744 . . . . . . . . . 10 (((𝑥 = 1o𝑦 = 2o) ∧ (𝑦 = 1o𝑧 = ∅)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
4746ex 413 . . . . . . . . 9 ((𝑥 = 1o𝑦 = 2o) → ((𝑦 = 1o𝑧 = ∅) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
4845ad2ant2lr 744 . . . . . . . . . 10 (((𝑥 = 1o𝑦 = 2o) ∧ (𝑦 = 1o𝑧 = 2o)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
4948ex 413 . . . . . . . . 9 ((𝑥 = 1o𝑦 = 2o) → ((𝑦 = 1o𝑧 = 2o) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
50 eqtr2 2846 . . . . . . . . . . . . 13 ((𝑦 = 2o𝑦 = ∅) → 2o = ∅)
5119, 50mto 198 . . . . . . . . . . . 12 ¬ (𝑦 = 2o𝑦 = ∅)
5251pm2.21i 119 . . . . . . . . . . 11 ((𝑦 = 2o𝑦 = ∅) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
5352ad2ant2lr 744 . . . . . . . . . 10 (((𝑥 = 1o𝑦 = 2o) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
5453ex 413 . . . . . . . . 9 ((𝑥 = 1o𝑦 = 2o) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
5547, 49, 543jaod 1422 . . . . . . . 8 ((𝑥 = 1o𝑦 = 2o) → (((𝑦 = 1o𝑧 = ∅) ∨ (𝑦 = 1o𝑧 = 2o) ∨ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
5645ad2ant2lr 744 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = 1o𝑧 = ∅)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
5756ex 413 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = 1o𝑧 = ∅) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
5845ad2ant2lr 744 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = 1o𝑧 = 2o)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
5958ex 413 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = 1o𝑧 = 2o) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
6052ad2ant2lr 744 . . . . . . . . . 10 (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
6160ex 413 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
6257, 59, 613jaod 1422 . . . . . . . 8 ((𝑥 = ∅ ∧ 𝑦 = 2o) → (((𝑦 = 1o𝑧 = ∅) ∨ (𝑦 = 1o𝑧 = 2o) ∨ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
6342, 55, 623jaoi 1421 . . . . . . 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 407 . . . . . 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 597 . . . . 5 ((𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑦𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧) → ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
6624, 30brtp 32869 . . . . 5 (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧 ↔ ((𝑥 = 1o𝑧 = ∅) ∨ (𝑥 = 1o𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
6765, 66sylibr 235 . . . 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 4624 . . . . 5 (𝑥 ∈ {1o, 2o, ∅} ↔ (𝑥 = 1o𝑥 = 2o𝑥 = ∅))
7028eltp 4624 . . . . 5 (𝑦 ∈ {1o, 2o, ∅} ↔ (𝑦 = 1o𝑦 = 2o𝑦 = ∅))
71 eqtr3 2847 . . . . . . . . . 10 ((𝑥 = 1o𝑦 = 1o) → 𝑥 = 𝑦)
72713mix2d 1331 . . . . . . . . 9 ((𝑥 = 1o𝑦 = 1o) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
7372ex 413 . . . . . . . 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 1325 . . . . . . . . . 10 ((𝑥 = 1o𝑦 = 2o) → ((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
75743mix1d 1330 . . . . . . . . 9 ((𝑥 = 1o𝑦 = 2o) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
7675ex 413 . . . . . . . 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 1324 . . . . . . . . . 10 ((𝑥 = 1o𝑦 = ∅) → ((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
78773mix1d 1330 . . . . . . . . 9 ((𝑥 = 1o𝑦 = ∅) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
7978ex 413 . . . . . . . 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 1422 . . . . . . 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 1325 . . . . . . . . . 10 ((𝑦 = 1o𝑥 = 2o) → ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
82813mix3d 1332 . . . . . . . . 9 ((𝑦 = 1o𝑥 = 2o) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
8382expcom 414 . . . . . . . 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 2847 . . . . . . . . . 10 ((𝑥 = 2o𝑦 = 2o) → 𝑥 = 𝑦)
85843mix2d 1331 . . . . . . . . 9 ((𝑥 = 2o𝑦 = 2o) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
8685ex 413 . . . . . . . 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 1326 . . . . . . . . . 10 ((𝑦 = ∅ ∧ 𝑥 = 2o) → ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
88873mix3d 1332 . . . . . . . . 9 ((𝑦 = ∅ ∧ 𝑥 = 2o) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
8988expcom 414 . . . . . . . 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 1422 . . . . . . 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 1324 . . . . . . . . . 10 ((𝑦 = 1o𝑥 = ∅) → ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
92913mix3d 1332 . . . . . . . . 9 ((𝑦 = 1o𝑥 = ∅) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
9392expcom 414 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = 1o → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))))
94 3mix3 1326 . . . . . . . . . 10 ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
95943mix1d 1330 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = 2o) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
9695ex 413 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = 2o → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))))
97 eqtr3 2847 . . . . . . . . . 10 ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑥 = 𝑦)
98973mix2d 1331 . . . . . . . . 9 ((𝑥 = ∅ ∧ 𝑦 = ∅) → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o))))
9998ex 413 . . . . . . . 8 (𝑥 = ∅ → (𝑦 = ∅ → (((𝑥 = 1o𝑦 = ∅) ∨ (𝑥 = 1o𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)) ∨ 𝑥 = 𝑦 ∨ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))))
10093, 96, 993jaod 1422 . . . . . . 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 1421 . . . . . 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 407 . . . . 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 597 . . . 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 262 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
10528, 24brtp 32869 . . . . 5 (𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥 ↔ ((𝑦 = 1o𝑥 = ∅) ∨ (𝑦 = 1o𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
10629, 104, 1053orbi123i 1150 . . . 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 235 . . 3 ((𝑥 ∈ {1o, 2o, ∅} ∧ 𝑦 ∈ {1o, 2o, ∅}) → (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑦𝑥 = 𝑦𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥))
10827, 68, 107issoi 5505 . 2 {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or {1o, 2o, ∅}
109 df-tp 4568 . . 3 {1o, 2o, ∅} = ({1o, 2o} ∪ {∅})
110 soeq2 5493 . . 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 231 1 {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1080   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ∪ cun 3937  ∅c0 4294  {csn 4563  {cpr 4565  {ctp 4567  ⟨cop 4569   class class class wbr 5062   Or wor 5471  Oncon0 6188  suc csuc 6190  1oc1o 8089  2oc2o 8090 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-ord 6191  df-on 6192  df-suc 6194  df-1o 8096  df-2o 8097 This theorem is referenced by:  sltso  33065
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