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Mirrors > Home > ILE Home > Th. List > exmidel | GIF version |
Description: Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.) |
Ref | Expression |
---|---|
exmidel | ⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidexmid 4128 | . . 3 ⊢ (EXMID → DECID 𝑥 ∈ 𝑦) | |
2 | 1 | alrimivv 1848 | . 2 ⊢ (EXMID → ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦) |
3 | 0ex 4063 | . . . 4 ⊢ ∅ ∈ V | |
4 | eleq1 2203 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦)) | |
5 | 4 | dcbid 824 | . . . . 5 ⊢ (𝑥 = ∅ → (DECID 𝑥 ∈ 𝑦 ↔ DECID ∅ ∈ 𝑦)) |
6 | 5 | albidv 1797 | . . . 4 ⊢ (𝑥 = ∅ → (∀𝑦DECID 𝑥 ∈ 𝑦 ↔ ∀𝑦DECID ∅ ∈ 𝑦)) |
7 | 3, 6 | spcv 2783 | . . 3 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → ∀𝑦DECID ∅ ∈ 𝑦) |
8 | exmid0el 4135 | . . 3 ⊢ (EXMID ↔ ∀𝑦DECID ∅ ∈ 𝑦) | |
9 | 7, 8 | sylibr 133 | . 2 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → EXMID) |
10 | 2, 9 | impbii 125 | 1 ⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 DECID wdc 820 ∀wal 1330 = wceq 1332 ∈ wcel 1481 ∅c0 3368 EXMIDwem 4126 |
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-pow 4106 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-exmid 4127 |
This theorem is referenced by: (None) |
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