Theorem exmid1stab 4237

Description: If every 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 4156	. . . . . . . . 9 ⊢ ∅ ∈ V
2	1	snid 3649	. . . . . . . 8 ⊢ ∅ ∈ {∅}
3		nnexmid 851	. . . . . . . . . . 11 ⊢ ¬ ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})
4		uneq1 3306	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = ({∅} ∪ ({∅} ∖ 𝑦)))
5		undifabs 3523	. . . . . . . . . . . . . . 15 ⊢ ({∅} ∪ ({∅} ∖ 𝑦)) = {∅}
6	4, 5	eqtrdi 2242	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
7	6	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ {∅} → (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
8		df-ne 2365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ≠ {∅} ↔ ¬ 𝑦 = {∅})
9		pwntru 4228	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ {∅} ∧ 𝑦 ≠ {∅}) → 𝑦 = ∅)
10	8, 9	sylan2br 288	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → 𝑦 = ∅)
11	10	difeq2d 3277	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = ({∅} ∖ ∅))
12		dif0 3517	. . . . . . . . . . . . . . . . 17 ⊢ ({∅} ∖ ∅) = {∅}
13	11, 12	eqtrdi 2242	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = {∅})
14	10, 13	uneq12d 3314	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = (∅ ∪ {∅}))
15		uncom 3303	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
16		un0 3480	. . . . . . . . . . . . . . . 16 ⊢ ({∅} ∪ ∅) = {∅}
17	15, 16	eqtri 2214	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {∅}) = {∅}
18	14, 17	eqtrdi 2242	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
19	18	ex 115	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ {∅} → (¬ 𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
20	7, 19	jaod 718	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ {∅} → ((𝑦 = {∅} ∨ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
21	20	con3d 632	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ {∅} → (¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})))
22	3, 21	mtoi 665	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
23	22	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
24		difss 3285	. . . . . . . . . . . . 13 ⊢ ({∅} ∖ 𝑦) ⊆ {∅}
25		unss 3333	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
26	25	biimpi 120	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
27	24, 26	mpan2 425	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
28	27	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
29		exmid1stab.x	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
30	29	ex 115	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
31	30	alrimiv 1885	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
32		p0ex 4217	. . . . . . . . . . . . . 14 ⊢ {∅} ∈ V
33	32	ssex 4166	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ∈ V)
34		sseq1 3202	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 ⊆ {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅}))
35		eqeq1 2200	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 = {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
36	35	stbid 833	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (STAB 𝑥 = {∅} ↔ STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
37	34, 36	imbi12d 234	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → ((𝑥 ⊆ {∅} → STAB 𝑥 = {∅}) ↔ ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})))
38	37	spcgv 2847	. . . . . . . . . . . . 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 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
41	28, 40	mpd 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
42		df-stab 832	. . . . . . . . . 10 ⊢ (STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} ↔ (¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
43	41, 42	sylib 122	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
44	23, 43	mpd 13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
45	2, 44	eleqtrrid 2283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → ∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)))
46		elun 3300	. . . . . . 7 ⊢ (∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)) ↔ (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
47	45, 46	sylib 122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
48		eldifn 3282	. . . . . . 7 ⊢ (∅ ∈ ({∅} ∖ 𝑦) → ¬ ∅ ∈ 𝑦)
49	48	orim2i 762	. . . . . 6 ⊢ ((∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
50	47, 49	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
51		df-dc 836	. . . . 5 ⊢ (DECID ∅ ∈ 𝑦 ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
52	50, 51	sylibr 134	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → DECID ∅ ∈ 𝑦)
53	52	ex 115	. . 3 ⊢ (𝜑 → (𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
54	53	alrimiv 1885	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
55		df-exmid 4224	. 2 ⊢ (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
56	54, 55	sylibr 134	1 ⊢ (𝜑 → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  STAB wstab 831  DECID wdc 835  ∀wal 1362   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  Vcvv 2760   ∖ cdif 3150   ∪ cun 3151   ⊆ wss 3153  ∅c0 3446  {csn 3618  EXMIDwem 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-exmid 4224

This theorem is referenced by:  2omotap  7319  subctctexmid  15491