![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 4063 | . . . . . . 7 ⊢ ∅ ∈ V | |
2 | 1 | bj-snex 13282 | . . . . . 6 ⊢ {∅} ∈ V |
3 | 2 | zfauscl 4056 | . . . . 5 ⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) |
4 | eleq1 2203 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑎 ↔ ∅ ∈ 𝑎)) | |
5 | eleq1 2203 | . . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∈ {∅} ↔ ∅ ∈ {∅})) | |
6 | 5 | anbi1d 461 | . . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ∈ {∅} ∧ 𝜑) ↔ (∅ ∈ {∅} ∧ 𝜑))) |
7 | 4, 6 | bibi12d 234 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)))) |
8 | 1, 7 | spcv 2783 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 ∈ {∅} ∧ 𝜑)) → (∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑))) |
9 | 3, 8 | eximii 1582 | . . . 4 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) |
10 | 1 | snid 3563 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
11 | 10 | biantrur 301 | . . . . . . 7 ⊢ (𝜑 ↔ (∅ ∈ {∅} ∧ 𝜑)) |
12 | 11 | bicomi 131 | . . . . . 6 ⊢ ((∅ ∈ {∅} ∧ 𝜑) ↔ 𝜑) |
13 | 12 | bibi2i 226 | . . . . 5 ⊢ ((∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ (∅ ∈ 𝑎 ↔ 𝜑)) |
14 | 13 | exbii 1585 | . . . 4 ⊢ (∃𝑎(∅ ∈ 𝑎 ↔ (∅ ∈ {∅} ∧ 𝜑)) ↔ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑)) |
15 | 9, 14 | mpbi 144 | . . 3 ⊢ ∃𝑎(∅ ∈ 𝑎 ↔ 𝜑) |
16 | bj-bd0el 13237 | . . . . 5 ⊢ BOUNDED ∅ ∈ 𝑎 | |
17 | 16 | ax-bj-d0cl 13293 | . . . 4 ⊢ DECID ∅ ∈ 𝑎 |
18 | dcbiit 825 | . . . 4 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → (DECID ∅ ∈ 𝑎 ↔ DECID 𝜑)) | |
19 | 17, 18 | mpbii 147 | . . 3 ⊢ ((∅ ∈ 𝑎 ↔ 𝜑) → DECID 𝜑) |
20 | 15, 19 | eximii 1582 | . 2 ⊢ ∃𝑎DECID 𝜑 |
21 | bj-ex 13140 | . 2 ⊢ (∃𝑎DECID 𝜑 → DECID 𝜑) | |
22 | 20, 21 | ax-mp 5 | 1 ⊢ DECID 𝜑 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 DECID wdc 820 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∅c0 3368 {csn 3532 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pr 4139 ax-bd0 13182 ax-bdim 13183 ax-bdor 13185 ax-bdn 13186 ax-bdal 13187 ax-bdex 13188 ax-bdeq 13189 ax-bdsep 13253 ax-bj-d0cl 13293 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-pr 3539 df-bdc 13210 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |