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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 3905 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | 1 | bj-snex 10704 | . . . . . 6 ⊢ {∅} ∈ V |
3 | 2 | zfauscl 3898 | . . . . 5 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) |
4 | eleq1 2141 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑎 ↔ ∅ ∈ 𝑎)) | |
5 | eleq1 2141 | . . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∈ {∅} ↔ ∅ ∈ {∅})) | |
6 | 5 | anbi1d 452 | . . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ∈ {∅} ∧ 𝜑) ↔ (∅ ∈ {∅} ∧ 𝜑))) |
7 | 4, 6 | bibi12d 233 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)))) |
8 | 1, 7 | spcv 2691 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) → (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑))) |
9 | 3, 8 | eximii 1533 | . . . 4 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) |
10 | 1 | snid 3425 | . . . . . . . 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 1536 | . . . 4 ⊢ (∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑)) |
15 | 9, 14 | mpbi 143 | . . 3 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑) |
16 | bj-bd0el 10659 | . . . . 5 ⊢ BOUNDED ∅ ∈ 𝑎 | |
17 | 16 | ax-bj-d0cl 10715 | . . . 4 ⊢ DECID ∅ ∈ 𝑎 |
18 | bj-dcbi 10719 | . . . 4 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → (DECID ∅ ∈ 𝑎 ↔ DECID 𝜑)) | |
19 | 17, 18 | mpbii 146 | . . 3 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → DECID 𝜑) |
20 | 15, 19 | eximii 1533 | . 2 ⊢ ∃𝑎DECID 𝜑 |
21 | bj-ex 10573 | . 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 775 ∀wal 1282 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∅c0 3251 {csn 3398 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pr 3964 ax-bd0 10604 ax-bdim 10605 ax-bdor 10607 ax-bdn 10608 ax-bdal 10609 ax-bdex 10610 ax-bdeq 10611 ax-bdsep 10675 ax-bj-d0cl 10715 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-pr 3405 df-bdc 10632 |
This theorem is referenced by: (None) |
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