Theorem exmid1stab 14033

Description: If any proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)

Hypothesis

Ref	Expression
exmid1stab.x	⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})

Assertion

Ref	Expression
exmid1stab	⊢ (𝜑 → EXMID)

Distinct variable group:   𝜑,𝑥

Proof of Theorem exmid1stab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4116	. . . . . . . . 9 ⊢ ∅ ∈ V
2	1	snid 3614	. . . . . . . 8 ⊢ ∅ ∈ {∅}
3		nnexmid 845	. . . . . . . . . . 11 ⊢ ¬ ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})
4		uneq1 3274	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = ({∅} ∪ ({∅} ∖ 𝑦)))
5		undifabs 3491	. . . . . . . . . . . . . . 15 ⊢ ({∅} ∪ ({∅} ∖ 𝑦)) = {∅}
6	4, 5	eqtrdi 2219	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
7	6	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ {∅} → (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
8		df-ne 2341	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ≠ {∅} ↔ ¬ 𝑦 = {∅})
9		pwntru 4185	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ {∅} ∧ 𝑦 ≠ {∅}) → 𝑦 = ∅)
10	8, 9	sylan2br 286	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → 𝑦 = ∅)
11	10	difeq2d 3245	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = ({∅} ∖ ∅))
12		dif0 3485	. . . . . . . . . . . . . . . . 17 ⊢ ({∅} ∖ ∅) = {∅}
13	11, 12	eqtrdi 2219	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = {∅})
14	10, 13	uneq12d 3282	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = (∅ ∪ {∅}))
15		uncom 3271	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
16		un0 3448	. . . . . . . . . . . . . . . 16 ⊢ ({∅} ∪ ∅) = {∅}
17	15, 16	eqtri 2191	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {∅}) = {∅}
18	14, 17	eqtrdi 2219	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
19	18	ex 114	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ {∅} → (¬ 𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
20	7, 19	jaod 712	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ {∅} → ((𝑦 = {∅} ∨ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
21	20	con3d 626	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ {∅} → (¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})))
22	3, 21	mtoi 659	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
23	22	adantl 275	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
24		difss 3253	. . . . . . . . . . . . 13 ⊢ ({∅} ∖ 𝑦) ⊆ {∅}
25		unss 3301	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
26	25	biimpi 119	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
27	24, 26	mpan2 423	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
28	27	adantl 275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
29		exmid1stab.x	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
30	29	ex 114	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
31	30	alrimiv 1867	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
32		p0ex 4174	. . . . . . . . . . . . . 14 ⊢ {∅} ∈ V
33	32	ssex 4126	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ∈ V)
34		sseq1 3170	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 ⊆ {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅}))
35		eqeq1 2177	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 = {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
36	35	stbid 827	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (STAB 𝑥 = {∅} ↔ STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
37	34, 36	imbi12d 233	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → ((𝑥 ⊆ {∅} → STAB 𝑥 = {∅}) ↔ ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})))
38	37	spcgv 2817	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∪ ({∅} ∖ 𝑦)) ∈ V → (∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}) → ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})))
39	27, 33, 38	3syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ {∅} → (∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}) → ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})))
40	31, 39	mpan9 279	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
41	28, 40	mpd 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
42		df-stab 826	. . . . . . . . . 10 ⊢ (STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} ↔ (¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
43	41, 42	sylib 121	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
44	23, 43	mpd 13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
45	2, 44	eleqtrrid 2260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → ∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)))
46		elun 3268	. . . . . . 7 ⊢ (∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)) ↔ (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
47	45, 46	sylib 121	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
48		eldifn 3250	. . . . . . 7 ⊢ (∅ ∈ ({∅} ∖ 𝑦) → ¬ ∅ ∈ 𝑦)
49	48	orim2i 756	. . . . . 6 ⊢ ((∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
50	47, 49	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
51		df-dc 830	. . . . 5 ⊢ (DECID ∅ ∈ 𝑦 ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
52	50, 51	sylibr 133	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → DECID ∅ ∈ 𝑦)
53	52	ex 114	. . 3 ⊢ (𝜑 → (𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
54	53	alrimiv 1867	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
55		df-exmid 4181	. 2 ⊢ (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
56	54, 55	sylibr 133	1 ⊢ (𝜑 → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703  STAB wstab 825  DECID wdc 829  ∀wal 1346   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  Vcvv 2730   ∖ cdif 3118   ∪ cun 3119   ⊆ wss 3121  ∅c0 3414  {csn 3583  EXMIDwem 4180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-exmid 4181

This theorem is referenced by:  subctctexmid  14034