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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 3884 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | 1 | bj-snex 10033 | . . . . . 6 ⊢ {∅} ∈ V |
3 | 2 | zfauscl 3877 | . . . . 5 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) |
4 | eleq1 2100 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑎 ↔ ∅ ∈ 𝑎)) | |
5 | eleq1 2100 | . . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∈ {∅} ↔ ∅ ∈ {∅})) | |
6 | 5 | anbi1d 438 | . . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ∈ {∅} ∧ 𝜑) ↔ (∅ ∈ {∅} ∧ 𝜑))) |
7 | 4, 6 | bibi12d 224 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)))) |
8 | 1, 7 | spcv 2646 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) → (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑))) |
9 | 3, 8 | eximii 1493 | . . . 4 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) |
10 | 1 | snid 3402 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
11 | 10 | biantrur 287 | . . . . . . 7 ⊢ (𝜑 ↔ (∅ ∈ {∅} ∧ 𝜑)) |
12 | 11 | bicomi 123 | . . . . . 6 ⊢ ((∅ ∈ {∅} ∧ 𝜑) ↔ 𝜑) |
13 | 12 | bibi2i 216 | . . . . 5 ⊢ ((∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ 𝜑)) |
14 | 13 | exbii 1496 | . . . 4 ⊢ (∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑)) |
15 | 9, 14 | mpbi 133 | . . 3 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑) |
16 | bj-bd0el 9988 | . . . . 5 ⊢ BOUNDED ∅ ∈ 𝑎 | |
17 | 16 | ax-bj-d0cl 10044 | . . . 4 ⊢ DECID ∅ ∈ 𝑎 |
18 | bj-dcbi 10048 | . . . 4 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → (DECID ∅ ∈ 𝑎 ↔ DECID 𝜑)) | |
19 | 17, 18 | mpbii 136 | . . 3 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → DECID 𝜑) |
20 | 15, 19 | eximii 1493 | . 2 ⊢ ∃𝑎DECID 𝜑 |
21 | bj-ex 9902 | . 2 ⊢ (∃𝑎DECID 𝜑 → DECID 𝜑) | |
22 | 20, 21 | ax-mp 7 | 1 ⊢ DECID 𝜑 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 DECID wdc 742 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∅c0 3224 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pr 3944 ax-bd0 9933 ax-bdim 9934 ax-bdor 9936 ax-bdn 9937 ax-bdal 9938 ax-bdex 9939 ax-bdeq 9940 ax-bdsep 10004 ax-bj-d0cl 10044 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-pr 3382 df-bdc 9961 |
This theorem is referenced by: (None) |
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