Theorem exmid1stab 4257

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 4176	. . . . . . . . 9 ⊢ ∅ ∈ V
2	1	snid 3666	. . . . . . . 8 ⊢ ∅ ∈ {∅}
3		nnexmid 852	. . . . . . . . . . 11 ⊢ ¬ ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})
4		uneq1 3322	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = ({∅} ∪ ({∅} ∖ 𝑦)))
5		undifabs 3539	. . . . . . . . . . . . . . 15 ⊢ ({∅} ∪ ({∅} ∖ 𝑦)) = {∅}
6	4, 5	eqtrdi 2255	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
7	6	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ {∅} → (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
8		df-ne 2378	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ≠ {∅} ↔ ¬ 𝑦 = {∅})
9		pwntru 4248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ {∅} ∧ 𝑦 ≠ {∅}) → 𝑦 = ∅)
10	8, 9	sylan2br 288	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → 𝑦 = ∅)
11	10	difeq2d 3293	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = ({∅} ∖ ∅))
12		dif0 3533	. . . . . . . . . . . . . . . . 17 ⊢ ({∅} ∖ ∅) = {∅}
13	11, 12	eqtrdi 2255	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = {∅})
14	10, 13	uneq12d 3330	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = (∅ ∪ {∅}))
15		uncom 3319	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∪ {∅}) = ({∅} ∪ ∅)
16		un0 3496	. . . . . . . . . . . . . . . 16 ⊢ ({∅} ∪ ∅) = {∅}
17	15, 16	eqtri 2227	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {∅}) = {∅}
18	14, 17	eqtrdi 2255	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
19	18	ex 115	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ {∅} → (¬ 𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
20	7, 19	jaod 719	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ {∅} → ((𝑦 = {∅} ∨ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
21	20	con3d 632	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ {∅} → (¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})))
22	3, 21	mtoi 666	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {∅} → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
23	22	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
24		difss 3301	. . . . . . . . . . . . 13 ⊢ ({∅} ∖ 𝑦) ⊆ {∅}
25		unss 3349	. . . . . . . . . . . . . 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 1898	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
32		p0ex 4237	. . . . . . . . . . . . . 14 ⊢ {∅} ∈ V
33	32	ssex 4186	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ∈ V)
34		sseq1 3218	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 ⊆ {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅}))
35		eqeq1 2213	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 = {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
36	35	stbid 834	. . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . 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 833	. . . . . . . . . 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 2296	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → ∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)))
46		elun 3316	. . . . . . 7 ⊢ (∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)) ↔ (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
47	45, 46	sylib 122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
48		eldifn 3298	. . . . . . 7 ⊢ (∅ ∈ ({∅} ∖ 𝑦) → ¬ ∅ ∈ 𝑦)
49	48	orim2i 763	. . . . . 6 ⊢ ((∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
50	47, 49	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
51		df-dc 837	. . . . 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 1898	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
55		df-exmid 4244	. 2 ⊢ (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
56	54, 55	sylibr 134	1 ⊢ (𝜑 → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710  STAB wstab 832  DECID wdc 836  ∀wal 1371   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  Vcvv 2773   ∖ cdif 3165   ∪ cun 3166   ⊆ wss 3168  ∅c0 3462  {csn 3635  EXMIDwem 4243

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-nul 4175  ax-pow 4223

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rab 2494  df-v 2775  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-exmid 4244

This theorem is referenced by:  2omotap  7384  subctctexmid  16052