Theorem ltssolem1 27876

Description: Lemma for ltsso 27877. 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 8481	. . . . . . . 8 ⊢ 1o ≠ ∅
2	1	neii 2963	. . . . . . 7 ⊢ ¬ 1o = ∅
3		eqtr2 2787	. . . . . . 7 ⊢ ((𝑥 = 1o ∧ 𝑥 = ∅) → 1o = ∅)
4	2, 3	mto 200	. . . . . 6 ⊢ ¬ (𝑥 = 1o ∧ 𝑥 = ∅)
5		1on 8475	. . . . . . . . 9 ⊢ 1o ∈ On
6		0elon 6423	. . . . . . . . 9 ⊢ ∅ ∈ On
7		df-2o 8463	. . . . . . . . . . 11 ⊢ 2o = suc 1o
8		df-1o 8462	. . . . . . . . . . 11 ⊢ 1o = suc ∅
9	7, 8	eqeq12i 2784	. . . . . . . . . 10 ⊢ (2o = 1o ↔ suc 1o = suc ∅)
10		suc11 6477	. . . . . . . . . 10 ⊢ ((1o ∈ On ∧ ∅ ∈ On) → (suc 1o = suc ∅ ↔ 1o = ∅))
11	9, 10	bitrid 286	. . . . . . . . 9 ⊢ ((1o ∈ On ∧ ∅ ∈ On) → (2o = 1o ↔ 1o = ∅))
12	5, 6, 11	mp2an 705	. . . . . . . 8 ⊢ (2o = 1o ↔ 1o = ∅)
13	1, 12	nemtbir 3057	. . . . . . 7 ⊢ ¬ 2o = 1o
14		eqtr2 2787	. . . . . . . 8 ⊢ ((𝑥 = 2o ∧ 𝑥 = 1o) → 2o = 1o)
15	14	ancoms 464	. . . . . . 7 ⊢ ((𝑥 = 1o ∧ 𝑥 = 2o) → 2o = 1o)
16	13, 15	mto 200	. . . . . 6 ⊢ ¬ (𝑥 = 1o ∧ 𝑥 = 2o)
17		nsuceq0 6453	. . . . . . . 8 ⊢ suc 1o ≠ ∅
18	7	eqeq1i 2771	. . . . . . . 8 ⊢ (2o = ∅ ↔ suc 1o = ∅)
19	17, 18	nemtbir 3057	. . . . . . 7 ⊢ ¬ 2o = ∅
20		eqtr2 2787	. . . . . . . 8 ⊢ ((𝑥 = 2o ∧ 𝑥 = ∅) → 2o = ∅)
21	20	ancoms 464	. . . . . . 7 ⊢ ((𝑥 = ∅ ∧ 𝑥 = 2o) → 2o = ∅)
22	19, 21	mto 200	. . . . . 6 ⊢ ¬ (𝑥 = ∅ ∧ 𝑥 = 2o)
23	4, 16, 22	3pm3.2ni 1519	. . . . 5 ⊢ ¬ ((𝑥 = 1o ∧ 𝑥 = ∅) ∨ (𝑥 = 1o ∧ 𝑥 = 2o) ∨ (𝑥 = ∅ ∧ 𝑥 = 2o))
24		vex 3462	. . . . . 6 ⊢ 𝑥 ∈ V
25	24, 24	brtp 5512	. . . . 5 ⊢ (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥 ↔ ((𝑥 = 1o ∧ 𝑥 = ∅) ∨ (𝑥 = 1o ∧ 𝑥 = 2o) ∨ (𝑥 = ∅ ∧ 𝑥 = 2o)))
26	23, 25	mtbir 326	. . . 4 ⊢ ¬ 𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥
27	26	a1i 11	. . 3 ⊢ (𝑥 ∈ {1o, 2o, ∅} → ¬ 𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥)
28		vex 3462	. . . . . . 7 ⊢ 𝑦 ∈ V
29	24, 28	brtp 5512	. . . . . 6 ⊢ (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑦 ↔ ((𝑥 = 1o ∧ 𝑦 = ∅) ∨ (𝑥 = 1o ∧ 𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
30		vex 3462	. . . . . . 7 ⊢ 𝑧 ∈ V
31	28, 30	brtp 5512	. . . . . 6 ⊢ (𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧 ↔ ((𝑦 = 1o ∧ 𝑧 = ∅) ∨ (𝑦 = 1o ∧ 𝑧 = 2o) ∨ (𝑦 = ∅ ∧ 𝑧 = 2o)))
32		eqtr2 2787	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 1o ∧ 𝑦 = ∅) → 1o = ∅)
33	2, 32	mto 200	. . . . . . . . . . . 12 ⊢ ¬ (𝑦 = 1o ∧ 𝑦 = ∅)
34	33	pm2.21i 120	. . . . . . . . . . 11 ⊢ ((𝑦 = 1o ∧ 𝑦 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
35	34	ad2ant2rl 762	. . . . . . . . . 10 ⊢ (((𝑦 = 1o ∧ 𝑧 = ∅) ∧ (𝑥 = 1o ∧ 𝑦 = ∅)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
36	35	expcom 419	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = ∅) → ((𝑦 = 1o ∧ 𝑧 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
37	34	ad2ant2rl 762	. . . . . . . . . 10 ⊢ (((𝑦 = 1o ∧ 𝑧 = 2o) ∧ (𝑥 = 1o ∧ 𝑦 = ∅)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
38	37	expcom 419	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = ∅) → ((𝑦 = 1o ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
39		3mix2 1350	. . . . . . . . . . 11 ⊢ ((𝑥 = 1o ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
40	39	ad2ant2rl 762	. . . . . . . . . 10 ⊢ (((𝑥 = 1o ∧ 𝑦 = ∅) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
41	40	ex 418	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = ∅) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
42	36, 38, 41	3jaod 1456	. . . . . . . 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 2787	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 2o ∧ 𝑦 = 1o) → 2o = 1o)
44	13, 43	mto 200	. . . . . . . . . . . 12 ⊢ ¬ (𝑦 = 2o ∧ 𝑦 = 1o)
45	44	pm2.21i 120	. . . . . . . . . . 11 ⊢ ((𝑦 = 2o ∧ 𝑦 = 1o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
46	45	ad2ant2lr 761	. . . . . . . . . 10 ⊢ (((𝑥 = 1o ∧ 𝑦 = 2o) ∧ (𝑦 = 1o ∧ 𝑧 = ∅)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
47	46	ex 418	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = 2o) → ((𝑦 = 1o ∧ 𝑧 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
48	45	ad2ant2lr 761	. . . . . . . . . 10 ⊢ (((𝑥 = 1o ∧ 𝑦 = 2o) ∧ (𝑦 = 1o ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
49	48	ex 418	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = 2o) → ((𝑦 = 1o ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
50		eqtr2 2787	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 2o ∧ 𝑦 = ∅) → 2o = ∅)
51	19, 50	mto 200	. . . . . . . . . . . 12 ⊢ ¬ (𝑦 = 2o ∧ 𝑦 = ∅)
52	51	pm2.21i 120	. . . . . . . . . . 11 ⊢ ((𝑦 = 2o ∧ 𝑦 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
53	52	ad2ant2lr 761	. . . . . . . . . 10 ⊢ (((𝑥 = 1o ∧ 𝑦 = 2o) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
54	53	ex 418	. . . . . . . . 9 ⊢ ((𝑥 = 1o ∧ 𝑦 = 2o) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
55	47, 49, 54	3jaod 1456	. . . . . . . 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 761	. . . . . . . . . 10 ⊢ (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = 1o ∧ 𝑧 = ∅)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
57	56	ex 418	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = 1o ∧ 𝑧 = ∅) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
58	45	ad2ant2lr 761	. . . . . . . . . 10 ⊢ (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = 1o ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
59	58	ex 418	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = 1o ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
60	52	ad2ant2lr 761	. . . . . . . . . 10 ⊢ (((𝑥 = ∅ ∧ 𝑦 = 2o) ∧ (𝑦 = ∅ ∧ 𝑧 = 2o)) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
61	60	ex 418	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑦 = ∅ ∧ 𝑧 = 2o) → ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o))))
62	57, 59, 61	3jaod 1456	. . . . . . . 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 1454	. . . . . . 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 412	. . . . . 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 610	. . . . 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 5512	. . . . 5 ⊢ (𝑥{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑧 ↔ ((𝑥 = 1o ∧ 𝑧 = ∅) ∨ (𝑥 = 1o ∧ 𝑧 = 2o) ∨ (𝑥 = ∅ ∧ 𝑧 = 2o)))
67	65, 66	sylibr 237	. . . 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 4660	. . . . 5 ⊢ (𝑥 ∈ {1o, 2o, ∅} ↔ (𝑥 = 1o ∨ 𝑥 = 2o ∨ 𝑥 = ∅))
70	28	eltp 4660	. . . . 5 ⊢ (𝑦 ∈ {1o, 2o, ∅} ↔ (𝑦 = 1o ∨ 𝑦 = 2o ∨ 𝑦 = ∅))
71		eqtr3 2788	. . . . . . . . . 10 ⊢ ((𝑥 = 1o ∧ 𝑦 = 1o) → 𝑥 = 𝑦)
72	71	3mix2d 1356	. . . . . . . . 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 418	. . . . . . . 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 1350	. . . . . . . . . 10 ⊢ ((𝑥 = 1o ∧ 𝑦 = 2o) → ((𝑥 = 1o ∧ 𝑦 = ∅) ∨ (𝑥 = 1o ∧ 𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
75	74	3mix1d 1355	. . . . . . . . 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 418	. . . . . . . 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 1349	. . . . . . . . . 10 ⊢ ((𝑥 = 1o ∧ 𝑦 = ∅) → ((𝑥 = 1o ∧ 𝑦 = ∅) ∨ (𝑥 = 1o ∧ 𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
78	77	3mix1d 1355	. . . . . . . . 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 418	. . . . . . . 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 1456	. . . . . . 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 1350	. . . . . . . . . 10 ⊢ ((𝑦 = 1o ∧ 𝑥 = 2o) → ((𝑦 = 1o ∧ 𝑥 = ∅) ∨ (𝑦 = 1o ∧ 𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
82	81	3mix3d 1357	. . . . . . . . 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 419	. . . . . . . 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 2788	. . . . . . . . . 10 ⊢ ((𝑥 = 2o ∧ 𝑦 = 2o) → 𝑥 = 𝑦)
85	84	3mix2d 1356	. . . . . . . . 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 418	. . . . . . . 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 1351	. . . . . . . . . 10 ⊢ ((𝑦 = ∅ ∧ 𝑥 = 2o) → ((𝑦 = 1o ∧ 𝑥 = ∅) ∨ (𝑦 = 1o ∧ 𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
88	87	3mix3d 1357	. . . . . . . . 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 419	. . . . . . . 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 1456	. . . . . . 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 1349	. . . . . . . . . 10 ⊢ ((𝑦 = 1o ∧ 𝑥 = ∅) → ((𝑦 = 1o ∧ 𝑥 = ∅) ∨ (𝑦 = 1o ∧ 𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
92	91	3mix3d 1357	. . . . . . . . 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 419	. . . . . . . 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 1351	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∧ 𝑦 = 2o) → ((𝑥 = 1o ∧ 𝑦 = ∅) ∨ (𝑥 = 1o ∧ 𝑦 = 2o) ∨ (𝑥 = ∅ ∧ 𝑦 = 2o)))
95	94	3mix1d 1355	. . . . . . . . 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 418	. . . . . . . 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 2788	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∧ 𝑦 = ∅) → 𝑥 = 𝑦)
98	97	3mix2d 1356	. . . . . . . . 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 418	. . . . . . . 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 1456	. . . . . . 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 1454	. . . . . 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 412	. . . . 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 610	. . . 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 264	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
105	28, 24	brtp 5512	. . . . 5 ⊢ (𝑦{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}𝑥 ↔ ((𝑦 = 1o ∧ 𝑥 = ∅) ∨ (𝑦 = 1o ∧ 𝑥 = 2o) ∨ (𝑦 = ∅ ∧ 𝑥 = 2o)))
106	29, 104, 105	3orbi123i 1174	. . . 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 237	. . 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 5610	. 2 ⊢ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or {1o, 2o, ∅}
109		df-tp 4599	. . 3 ⊢ {1o, 2o, ∅} = ({1o, 2o} ∪ {∅})
110		soeq2 5596	. . 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 233	1 ⊢ {⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} Or ({1o, 2o} ∪ {∅})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∪ cun 3906  ∅c0 4289  {csn 4594  {cpr 4596  {ctp 4598  ⟨cop 4600   class class class wbr 5114   Or wor 5573  Oncon0 6367  suc csuc 6369  1_oc1o 8455  2_oc2o 8456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6370  df-on 6371  df-suc 6373  df-1o 8462  df-2o 8463

This theorem is used by:  ltsso  27877