Theorem ltssolem1 27716

Description: Lemma for ltsso 27717. The "sign expansion" binary relation totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)

Assertion

Ref	Expression
ltssolem1	⊢ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅})

Proof of Theorem ltssolem1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 8451	. . . . . . . 8 ⊢ 1o ≠ ∅
2	1	neii 2958	. . . . . . 7 ⊢ ¬ 1o = ∅
3		eqtr2 2782	. . . . . . 7 ⊢ ((𝑥 = 1o ∧ 𝑥 = ∅) → 1o = ∅)
4	2, 3	mto 199	. . . . . 6 ⊢ ¬ (𝑥 = 1o ∧ 𝑥 = ∅)
5		1on 8445	. . . . . . . . 9 ⊢ 1o ∈ On
6		0elon 6397	. . . . . . . . 9 ⊢ ∅ ∈ On
7		df-2o 8433	. . . . . . . . . . 11 ⊢ 2o = suc 1o
8		df-1o 8432	. . . . . . . . . . 11 ⊢ 1o = suc ∅
9	7, 8	eqeq12i 2779	. . . . . . . . . 10 ⊢ (2o = 1o ↔ suc 1o = suc ∅)
10		suc11 6451	. . . . . . . . . 10 ⊢ ((1o ∈ On ∧ ∅ ∈ On) → (suc 1o = suc ∅ ↔ 1o = ∅))
11	9, 10	bitrid 285	. . . . . . . . 9 ⊢ ((1o ∈ On ∧ ∅ ∈ On) → (2o = 1o ↔ 1o = ∅))
12	5, 6, 11	mp2an 702	. . . . . . . 8 ⊢ (2o = 1o ↔ 1o = ∅)
13	1, 12	nemtbir 3052	. . . . . . 7 ⊢ ¬ 2o = 1o
14		eqtr2 2782	. . . . . . . 8 ⊢ ((𝑥 = 2o ∧ 𝑥 = 1o) → 2o = 1o)
15	14	ancoms 462	. . . . . . 7 ⊢ ((𝑥 = 1o ∧ 𝑥 = 2o) → 2o = 1o)
16	13, 15	mto 199	. . . . . 6 ⊢ ¬ (𝑥 = 1o ∧ 𝑥 = 2o)
17		nsuceq0 6427	. . . . . . . 8 ⊢ suc 1o ≠ ∅
18	7	eqeq1i 2766	. . . . . . . 8 ⊢ (2o = ∅ ↔ suc 1o = ∅)
19	17, 18	nemtbir 3052	. . . . . . 7 ⊢ ¬ 2o = ∅
20		eqtr2 2782	. . . . . . . 8 ⊢ ((𝑥 = 2o ∧ 𝑥 = ∅) → 2o = ∅)
21	20	ancoms 462	. . . . . . 7 ⊢ ((𝑥 = ∅ ∧ 𝑥 = 2o) → 2o = ∅)
22	19, 21	mto 199	. . . . . 6 ⊢ ¬ (𝑥 = ∅ ∧ 𝑥 = 2o)
23	4, 16, 22	3pm3.2ni 1508	. . . . 5 ⊢ ¬ ((𝑥 = 1o ∧ 𝑥 = ∅) ∨ (𝑥 = 1o ∧ 𝑥 = 2o) ∨ (𝑥 = ∅ ∧ 𝑥 = 2o))
24		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
25	24, 24	brtp 5492	. . . . 5 ⊢ (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥 ↔ ((𝑥 = 1o ∧ 𝑥 = ∅) ∨ (𝑥 = 1o ∧ 𝑥 = 2o) ∨ (𝑥 = ∅ ∧ 𝑥 = 2o)))
26	23, 25	mtbir 325	. . . 4 ⊢ ¬ 𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥
27	26	a1i 11	. . 3 ⊢ (𝑥 ∈ {1o, 2o, ∅} → ¬ 𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥)
28		vex 3457	. . . . . . 7 ⊢ 𝑦 ∈ V
29	24, 28	brtp 5492	. . . . . 6 ⊢ (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑦 ↔ ((𝑥 = 1o ∧ 𝑦 = ∅) ∨ (𝑥 = 1o ∧ 𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
30		vex 3457	. . . . . . 7 ⊢ 𝑧 ∈ V
31	28, 30	brtp 5492	. . . . . 6 ⊢ (𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧 ↔ ((𝑦 = 1o ∧ 𝑧 = ∅) ∨ (𝑦 = 1o ∧ 𝑧 = 2o) ∨ (𝑦 = ∅ ∧ 𝑧 = 2o)))
32		eqtr2 2782	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 1o ∧ 𝑦 = ∅) → 1o = ∅)
33	2, 32	mto 199	. . . . . . . . . . . 12 ⊢ ¬ (𝑦 = 1o ∧ 𝑦 = ∅)
34	33	pm2.21i 119	. . . . . . . . . . 11 ⊢ ((𝑦 = 1o ∧ 𝑦 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
35	34	ad2ant2rl 759	. . . . . . . . . 10 ⊢ (((𝑦 = 1o ∧ 𝑧 = ∅) ∧ (𝑥 = 1o ∧ 𝑦 = ∅)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
36	35	expcom 417	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = ∅) → ((𝑦 = 1o ∧ 𝑧 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
37	34	ad2ant2rl 759	. . . . . . . . . 10 ⊢ (((𝑦 = 1o ∧ 𝑧 = 2o) ∧ (𝑥 = 1o ∧ 𝑦 = ∅)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
38	37	expcom 417	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = ∅) → ((𝑦 = 1o ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
39		3mix2 1344	. . . . . . . . . . 11 ⊢ ((𝑥 = 1o ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
40	39	ad2ant2rl 759	. . . . . . . . . 10 ⊢ (((𝑥 = 1o ∧ 𝑦 = ∅) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
41	40	ex 416	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = ∅) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
42	36, 38, 41	3jaod 1448	. . . . . . . 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 2782	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 2o ∧ 𝑦 = 1o) → 2o = 1o)
44	13, 43	mto 199	. . . . . . . . . . . 12 ⊢ ¬ (𝑦 = 2o ∧ 𝑦 = 1o)
45	44	pm2.21i 119	. . . . . . . . . . 11 ⊢ ((𝑦 = 2o ∧ 𝑦 = 1o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
46	45	ad2ant2lr 758	. . . . . . . . . 10 ⊢ (((𝑥 = 1o ∧ 𝑦 = 2o) ∧ (𝑦 = 1o ∧ 𝑧 = ∅)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
47	46	ex 416	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = 2o) → ((𝑦 = 1o ∧ 𝑧 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
48	45	ad2ant2lr 758	. . . . . . . . . 10 ⊢ (((𝑥 = 1o ∧ 𝑦 = 2o) ∧ (𝑦 = 1o ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
49	48	ex 416	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = 2o) → ((𝑦 = 1o ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
50		eqtr2 2782	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 2o ∧ 𝑦 = ∅) → 2o = ∅)
51	19, 50	mto 199	. . . . . . . . . . . 12 ⊢ ¬ (𝑦 = 2o ∧ 𝑦 = ∅)
52	51	pm2.21i 119	. . . . . . . . . . 11 ⊢ ((𝑦 = 2o ∧ 𝑦 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
53	52	ad2ant2lr 758	. . . . . . . . . 10 ⊢ (((𝑥 = 1o ∧ 𝑦 = 2o) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
54	53	ex 416	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = 2o) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
55	47, 49, 54	3jaod 1448	. . . . . . . 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))))
56	45	ad2ant2lr 758	. . . . . . . . . 10 ⊢ (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = 1o ∧ 𝑧 = ∅)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
57	56	ex 416	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = 1o ∧ 𝑧 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
58	45	ad2ant2lr 758	. . . . . . . . . 10 ⊢ (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = 1o ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
59	58	ex 416	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = 1o ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
60	52	ad2ant2lr 758	. . . . . . . . . 10 ⊢ (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
61	60	ex 416	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
62	57, 59, 61	3jaod 1448	. . . . . . . 8 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → (((𝑦 = 1o ∧ 𝑧 = ∅) ∨ (𝑦 = 1o ∧ 𝑧 = 2o) ∨ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
63	42, 55, 62	3jaoi 1446	. . . . . . 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))))
64	63	imp 410	. . . . . 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)))
65	29, 31, 64	syl2anb 607	. . . . 5 ⊢ ((𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑦 ∧ 𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
66	24, 30	brtp 5492	. . . . 5 ⊢ (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧 ↔ ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
67	65, 66	sylibr 236	. . . 4 ⊢ ((𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑦 ∧ 𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧) → 𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧)
68	67	a1i 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))
69	24	eltp 4647	. . . . 5 ⊢ (𝑥 ∈ {1o, 2o, ∅} ↔ (𝑥 = 1o ∨ 𝑥 = 2o ∨ 𝑥 = ∅))
70	28	eltp 4647	. . . . 5 ⊢ (𝑦 ∈ {1o, 2o, ∅} ↔ (𝑦 = 1o ∨ 𝑦 = 2o ∨ 𝑦 = ∅))
71		eqtr3 2783	. . . . . . . . . 10 ⊢ ((𝑥 = 1o ∧ 𝑦 = 1o) → 𝑥 = 𝑦)
72	71	3mix2d 1350	. . . . . . . . 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))))
73	72	ex 416	. . . . . . . 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 1344	. . . . . . . . . 10 ⊢ ((𝑥 = 1o ∧ 𝑦 = 2o) → ((𝑥 = 1o ∧ 𝑦 = ∅) ∨ (𝑥 = 1o ∧ 𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
75	74	3mix1d 1349	. . . . . . . . 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))))
76	75	ex 416	. . . . . . . 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 1343	. . . . . . . . . 10 ⊢ ((𝑥 = 1o ∧ 𝑦 = ∅) → ((𝑥 = 1o ∧ 𝑦 = ∅) ∨ (𝑥 = 1o ∧ 𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
78	77	3mix1d 1349	. . . . . . . . 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))))
79	78	ex 416	. . . . . . . 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)))))
80	73, 76, 79	3jaod 1448	. . . . . . 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 1344	. . . . . . . . . 10 ⊢ ((𝑦 = 1o ∧ 𝑥 = 2o) → ((𝑦 = 1o ∧ 𝑥 = ∅) ∨ (𝑦 = 1o ∧ 𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
82	81	3mix3d 1351	. . . . . . . . 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))))
83	82	expcom 417	. . . . . . . 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 2783	. . . . . . . . . 10 ⊢ ((𝑥 = 2o ∧ 𝑦 = 2o) → 𝑥 = 𝑦)
85	84	3mix2d 1350	. . . . . . . . 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))))
86	85	ex 416	. . . . . . . 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 1345	. . . . . . . . . 10 ⊢ ((𝑦 = ∅ ∧ 𝑥 = 2o) → ((𝑦 = 1o ∧ 𝑥 = ∅) ∨ (𝑦 = 1o ∧ 𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
88	87	3mix3d 1351	. . . . . . . . 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))))
89	88	expcom 417	. . . . . . . 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)))))
90	83, 86, 89	3jaod 1448	. . . . . . 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 1343	. . . . . . . . . 10 ⊢ ((𝑦 = 1o ∧ 𝑥 = ∅) → ((𝑦 = 1o ∧ 𝑥 = ∅) ∨ (𝑦 = 1o ∧ 𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
92	91	3mix3d 1351	. . . . . . . . 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))))
93	92	expcom 417	. . . . . . . 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 1345	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑥 = 1o ∧ 𝑦 = ∅) ∨ (𝑥 = 1o ∧ 𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
95	94	3mix1d 1349	. . . . . . . . 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))))
96	95	ex 416	. . . . . . . 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 2783	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑥 = 𝑦)
98	97	3mix2d 1350	. . . . . . . . 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))))
99	98	ex 416	. . . . . . . 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)))))
100	93, 96, 99	3jaod 1448	. . . . . . 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)))))
101	80, 90, 100	3jaoi 1446	. . . . . 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)))))
102	101	imp 410	. . . . 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))))
103	69, 70, 102	syl2anb 607	. . . 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 263	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
105	28, 24	brtp 5492	. . . . 5 ⊢ (𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥 ↔ ((𝑦 = 1o ∧ 𝑥 = ∅) ∨ (𝑦 = 1o ∧ 𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
106	29, 104, 105	3orbi123i 1168	. . . 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))))
107	103, 106	sylibr 236	. . 3 ⊢ ((𝑥 ∈ {1o, 2o, ∅} ∧ 𝑦 ∈ {1o, 2o, ∅}) → (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥))
108	27, 68, 107	issoi 5589	. 2 ⊢ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or {1o, 2o, ∅}
109		df-tp 4586	. . 3 ⊢ {1o, 2o, ∅} = ({1o, 2o} ∪ {∅})
110		soeq2 5575	. . 3 ⊢ ({1o, 2o, ∅} = ({1o, 2o} ∪ {∅}) → ({⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or {1o, 2o, ∅} ↔ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅})))
111	109, 110	ax-mp 5	. 2 ⊢ ({⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or {1o, 2o, ∅} ↔ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅}))
112	108, 111	mpbi 232	1 ⊢ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ w3o 1096   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ∪ cun 3902  ∅c0 4285  {csn 4581  {cpr 4583  {ctp 4585  ⟨cop 4587   class class class wbr 5099   Or wor 5552  Oncon0 6342  suc csuc 6344  1_oc1o 8425  2_oc2o 8426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-ord 6345  df-on 6346  df-suc 6348  df-1o 8432  df-2o 8433

This theorem is referenced by:  ltsso  27717