Theorem bj-d0clsepcl 16624

Description: Δ₀-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) New usage is discouraged since this statement is not intuitionnistic. (New usage is discouraged.)

Assertion

Ref	Expression
bj-d0clsepcl	⊢ DECID 𝜑

Proof of Theorem bj-d0clsepcl

Dummy variables 𝑥 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4221	. . . . . . 7 ⊢ ∅ ∈ V
2	1	bj-snex 16612	. . . . . 6 ⊢ {∅} ∈ V
3	2	zfauscl 4214	. . . . 5 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑))
4		eleq1 2294	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑎 ↔ ∅ ∈ 𝑎))
5		eleq1 2294	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∈ {∅} ↔ ∅ ∈ {∅}))
6	5	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ∈ {∅} ∧ 𝜑) ↔ (∅ ∈ {∅} ∧ 𝜑)))
7	4, 6	bibi12d 235	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑))))
8	1, 7	spcv 2901	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) → (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)))
9	3, 8	eximii 1651	. . . 4 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑))
10	1	snid 3704	. . . . . . . 8 ⊢ ∅ ∈ {∅}
11	10	biantrur 303	. . . . . . 7 ⊢ (𝜑 ↔ (∅ ∈ {∅} ∧ 𝜑))
12	11	bicomi 132	. . . . . 6 ⊢ ((∅ ∈ {∅} ∧ 𝜑) ↔ 𝜑)
13	12	bibi2i 227	. . . . 5 ⊢ ((∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ 𝜑))
14	13	exbii 1654	. . . 4 ⊢ (∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑))
15	9, 14	mpbi 145	. . 3 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑)
16		bj-bd0el 16567	. . . . 5 ⊢ BOUNDED ∅ ∈ 𝑎
17	16	ax-bj-d0cl 16623	. . . 4 ⊢ DECID ∅ ∈ 𝑎
18		dcbiit 847	. . . 4 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → (DECID ∅ ∈ 𝑎 ↔ DECID 𝜑))
19	17, 18	mpbii 148	. . 3 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → DECID 𝜑)
20	15, 19	eximii 1651	. 2 ⊢ ∃𝑎DECID 𝜑
21		bj-ex 16463	. 2 ⊢ (∃𝑎DECID 𝜑 → DECID 𝜑)
22	20, 21	ax-mp 5	1 ⊢ DECID 𝜑

Syntax hints:   ∧ wa 104   ↔ wb 105  DECID wdc 842  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∅c0 3496  {csn 3673

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pr 4305  ax-bd0 16512  ax-bdim 16513  ax-bdor 16515  ax-bdn 16516  ax-bdal 16517  ax-bdex 16518  ax-bdeq 16519  ax-bdsep 16583  ax-bj-d0cl 16623

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-pr 3680  df-bdc 16540

This theorem is referenced by: (None)