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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-d0clsepcl | GIF version |
Description: Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-d0clsepcl | ⊢ DECID 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3966 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | 1 | bj-snex 11804 | . . . . . 6 ⊢ {∅} ∈ V |
3 | 2 | zfauscl 3959 | . . . . 5 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) |
4 | eleq1 2150 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑎 ↔ ∅ ∈ 𝑎)) | |
5 | eleq1 2150 | . . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∈ {∅} ↔ ∅ ∈ {∅})) | |
6 | 5 | anbi1d 453 | . . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ∈ {∅} ∧ 𝜑) ↔ (∅ ∈ {∅} ∧ 𝜑))) |
7 | 4, 6 | bibi12d 233 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)))) |
8 | 1, 7 | spcv 2712 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) → (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑))) |
9 | 3, 8 | eximii 1538 | . . . 4 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) |
10 | 1 | snid 3475 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
11 | 10 | biantrur 297 | . . . . . . 7 ⊢ (𝜑 ↔ (∅ ∈ {∅} ∧ 𝜑)) |
12 | 11 | bicomi 130 | . . . . . 6 ⊢ ((∅ ∈ {∅} ∧ 𝜑) ↔ 𝜑) |
13 | 12 | bibi2i 225 | . . . . 5 ⊢ ((∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ 𝜑)) |
14 | 13 | exbii 1541 | . . . 4 ⊢ (∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑)) |
15 | 9, 14 | mpbi 143 | . . 3 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑) |
16 | bj-bd0el 11759 | . . . . 5 ⊢ BOUNDED ∅ ∈ 𝑎 | |
17 | 16 | ax-bj-d0cl 11815 | . . . 4 ⊢ DECID ∅ ∈ 𝑎 |
18 | bj-dcbi 11819 | . . . 4 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → (DECID ∅ ∈ 𝑎 ↔ DECID 𝜑)) | |
19 | 17, 18 | mpbii 146 | . . 3 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → DECID 𝜑) |
20 | 15, 19 | eximii 1538 | . 2 ⊢ ∃𝑎DECID 𝜑 |
21 | bj-ex 11663 | . 2 ⊢ (∃𝑎DECID 𝜑 → DECID 𝜑) | |
22 | 20, 21 | ax-mp 7 | 1 ⊢ DECID 𝜑 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 DECID wdc 780 ∀wal 1287 = wceq 1289 ∃wex 1426 ∈ wcel 1438 ∅c0 3286 {csn 3446 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pr 4036 ax-bd0 11704 ax-bdim 11705 ax-bdor 11707 ax-bdn 11708 ax-bdal 11709 ax-bdex 11710 ax-bdeq 11711 ax-bdsep 11775 ax-bj-d0cl 11815 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-sn 3452 df-pr 3453 df-bdc 11732 |
This theorem is referenced by: (None) |
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