Theorem exmidsssnc 4189

Description: Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4184 but lets you choose any set as the element of the singleton rather than just ∅. It is similar to exmidsssn 4188 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 4188	. . . 4 ⊢ (EXMID ↔ ∀𝑥∀𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})))
2		sneq 3594	. . . . . . . 8 ⊢ (𝑢 = 𝐵 → {𝑢} = {𝐵})
3	2	sseq2d 3177	. . . . . . 7 ⊢ (𝑢 = 𝐵 → (𝑥 ⊆ {𝑢} ↔ 𝑥 ⊆ {𝐵}))
4	2	eqeq2d 2182	. . . . . . . 8 ⊢ (𝑢 = 𝐵 → (𝑥 = {𝑢} ↔ 𝑥 = {𝐵}))
5	4	orbi2d 785	. . . . . . 7 ⊢ (𝑢 = 𝐵 → ((𝑥 = ∅ ∨ 𝑥 = {𝑢}) ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵})))
6	3, 5	bibi12d 234	. . . . . 6 ⊢ (𝑢 = 𝐵 → ((𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) ↔ (𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
7	6	spcgv 2817	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∀𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) → (𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
8	7	alimdv 1872	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥∀𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) → ∀𝑥(𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
9	1, 8	syl5bi 151	. . 3 ⊢ (𝐵 ∈ 𝑉 → (EXMID → ∀𝑥(𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
10		biimp 117	. . . 4 ⊢ ((𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})))
11	10	alimi 1448	. . 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 3232	. . . . . . . . 9 ⊢ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵}
14		snexg 4170	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → {𝐵} ∈ V)
15		rabexg 4132	. . . . . . . . . 10 ⊢ ({𝐵} ∈ V → {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ∈ V)
16		sseq1 3170	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 ⊆ {𝐵} ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵}))
17		eqeq1 2177	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 = ∅ ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅))
18		eqeq1 2177	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 = {𝐵} ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))
19	17, 18	orbi12d 788	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ((𝑥 = ∅ ∨ 𝑥 = {𝐵}) ↔ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵})))
20	16, 19	imbi12d 233	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ((𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) ↔ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵} → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))))
21	20	spcgv 2817	. . . . . . . . . 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 3444	. . . . . . . . . . 11 ⊢ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦)
25		snmg 3701	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑉 → ∃𝑤 𝑤 ∈ {𝐵})
26		r19.3rmv 3505	. . . . . . . . . . . 12 ⊢ (∃𝑤 𝑤 ∈ {𝐵} → (¬ ∅ ∈ 𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦))
27	25, 26	syl 14	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑉 → (¬ ∅ ∈ 𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦))
28	24, 27	bitr4id 198	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ↔ ¬ ∅ ∈ 𝑦))
29	28	biimpd 143	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ → ¬ ∅ ∈ 𝑦))
30		snidg 3612	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
31	30	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → 𝐵 ∈ {𝐵})
32		simpr 109	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵})
33	31, 32	eleqtrrd 2250	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → 𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦})
34		biidd 171	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐵 → (∅ ∈ 𝑦 ↔ ∅ ∈ 𝑦))
35	34	elrab 2886	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ↔ (𝐵 ∈ {𝐵} ∧ ∅ ∈ 𝑦))
36	35	simprbi 273	. . . . . . . . . . 11 ⊢ (𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ∅ ∈ 𝑦)
37	33, 36	syl 14	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → ∅ ∈ 𝑦)
38	37	ex 114	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵} → ∅ ∈ 𝑦))
39	29, 38	orim12d 781	. . . . . . . 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 723	. . . . . . 7 ⊢ ((¬ ∅ ∈ 𝑦 ∨ ∅ ∈ 𝑦) ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
42	40, 41	syl6ib 160	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦)))
43		df-dc 830	. . . . . 6 ⊢ (DECID ∅ ∈ 𝑦 ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
44	42, 43	syl6ibr 161	. . . . 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 1869	. . 3 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦)))
47		df-exmid 4181	. . 3 ⊢ (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
48	46, 47	syl6ibr 161	. 2 ⊢ (𝐵 ∈ 𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → EXMID))
49	12, 48	impbid 128	1 ⊢ (𝐵 ∈ 𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  {crab 2452  Vcvv 2730   ⊆ 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-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-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-exmid 4181

This theorem is referenced by:  exmidunben  12381