Theorem exmidsssnc 4335

Description: Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4330 but lets you choose any set as the element of the singleton rather than just ∅. It is similar to exmidsssn 4334 but for a particular set 𝐵 rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)

Assertion

Ref	Expression
exmidsssnc	⊢ (𝐵 ∈ 𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑉

Proof of Theorem exmidsssnc

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

Step	Hyp	Ref	Expression
1		exmidsssn 4334	. . . 4 ⊢ (EXMID ↔ ∀𝑥∀𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})))
2		sneq 3716	. . . . . . . 8 ⊢ (𝑢 = 𝐵 → {𝑢} = {𝐵})
3	2	sseq2d 3278	. . . . . . 7 ⊢ (𝑢 = 𝐵 → (𝑥 ⊆ {𝑢} ↔ 𝑥 ⊆ {𝐵}))
4	2	eqeq2d 2250	. . . . . . . 8 ⊢ (𝑢 = 𝐵 → (𝑥 = {𝑢} ↔ 𝑥 = {𝐵}))
5	4	orbi2d 802	. . . . . . 7 ⊢ (𝑢 = 𝐵 → ((𝑥 = ∅ ∨ 𝑥 = {𝑢}) ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵})))
6	3, 5	bibi12d 235	. . . . . 6 ⊢ (𝑢 = 𝐵 → ((𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) ↔ (𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
7	6	spcgv 2912	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∀𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) → (𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
8	7	alimdv 1932	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥∀𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) → ∀𝑥(𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
9	1, 8	biimtrid 152	. . 3 ⊢ (𝐵 ∈ 𝑉 → (EXMID → ∀𝑥(𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
10		biimp 118	. . . 4 ⊢ ((𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})))
11	10	alimi 1508	. . 3 ⊢ (∀𝑥(𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})))
12	9, 11	syl6 33	. 2 ⊢ (𝐵 ∈ 𝑉 → (EXMID → ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
13		ssrab2 3333	. . . . . . . . 9 ⊢ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵}
14		snexg 4316	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → {𝐵} ∈ V)
15		rabexg 4274	. . . . . . . . . 10 ⊢ ({𝐵} ∈ V → {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ∈ V)
16		sseq1 3271	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 ⊆ {𝐵} ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵}))
17		eqeq1 2245	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 = ∅ ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅))
18		eqeq1 2245	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 = {𝐵} ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))
19	17, 18	orbi12d 805	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ((𝑥 = ∅ ∨ 𝑥 = {𝐵}) ↔ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵})))
20	16, 19	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ((𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) ↔ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵} → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))))
21	20	spcgv 2912	. . . . . . . . . 10 ⊢ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ∈ V → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵} → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))))
22	14, 15, 21	3syl 17	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵} → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))))
23	13, 22	mpii 44	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵})))
24		rabeq0 3552	. . . . . . . . . . 11 ⊢ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦)
25		snmg 3826	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑉 → ∃𝑤 𝑤 ∈ {𝐵})
26		r19.3rmv 3615	. . . . . . . . . . . 12 ⊢ (∃𝑤 𝑤 ∈ {𝐵} → (¬ ∅ ∈ 𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦))
27	25, 26	syl 14	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑉 → (¬ ∅ ∈ 𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦))
28	24, 27	bitr4id 199	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ↔ ¬ ∅ ∈ 𝑦))
29	28	biimpd 144	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ → ¬ ∅ ∈ 𝑦))
30		snidg 3734	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
31	30	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → 𝐵 ∈ {𝐵})
32		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵})
33	31, 32	eleqtrrd 2318	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → 𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦})
34		biidd 172	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐵 → (∅ ∈ 𝑦 ↔ ∅ ∈ 𝑦))
35	34	elrab 2982	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ↔ (𝐵 ∈ {𝐵} ∧ ∅ ∈ 𝑦))
36	35	simprbi 275	. . . . . . . . . . 11 ⊢ (𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ∅ ∈ 𝑦)
37	33, 36	syl 14	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → ∅ ∈ 𝑦)
38	37	ex 115	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵} → ∅ ∈ 𝑦))
39	29, 38	orim12d 798	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → (¬ ∅ ∈ 𝑦 ∨ ∅ ∈ 𝑦)))
40	23, 39	syld 45	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (¬ ∅ ∈ 𝑦 ∨ ∅ ∈ 𝑦)))
41		orcom 740	. . . . . . 7 ⊢ ((¬ ∅ ∈ 𝑦 ∨ ∅ ∈ 𝑦) ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
42	40, 41	imbitrdi 161	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦)))
43		df-dc 847	. . . . . 6 ⊢ (DECID ∅ ∈ 𝑦 ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
44	42, 43	imbitrrdi 162	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → DECID ∅ ∈ 𝑦))
45	44	a1dd 48	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦)))
46	45	alrimdv 1929	. . 3 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦)))
47		df-exmid 4327	. . 3 ⊢ (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
48	46, 47	imbitrrdi 162	. 2 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → EXMID))
49	12, 48	impbid 129	1 ⊢ (𝐵 ∈ 𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ⊆ wss 3220  ∅c0 3520  {csn 3705  EXMIDwem 4326

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-exmid 4327

This theorem is referenced by:  exmidunben  13295