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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 4196 | . . 3 ⊢ (EXMID → DECID 𝑥 ∈ 𝑦) | |
2 | 1 | alrimivv 1875 | . 2 ⊢ (EXMID → ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦) |
3 | 0ex 4130 | . . . 4 ⊢ ∅ ∈ V | |
4 | eleq1 2240 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦)) | |
5 | 4 | dcbid 838 | . . . . 5 ⊢ (𝑥 = ∅ → (DECID 𝑥 ∈ 𝑦 ↔ DECID ∅ ∈ 𝑦)) |
6 | 5 | albidv 1824 | . . . 4 ⊢ (𝑥 = ∅ → (∀𝑦DECID 𝑥 ∈ 𝑦 ↔ ∀𝑦DECID ∅ ∈ 𝑦)) |
7 | 3, 6 | spcv 2831 | . . 3 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → ∀𝑦DECID ∅ ∈ 𝑦) |
8 | exmid0el 4204 | . . 3 ⊢ (EXMID ↔ ∀𝑦DECID ∅ ∈ 𝑦) | |
9 | 7, 8 | sylibr 134 | . 2 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → EXMID) |
10 | 2, 9 | impbii 126 | 1 ⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 DECID wdc 834 ∀wal 1351 = wceq 1353 ∈ wcel 2148 ∅c0 3422 EXMIDwem 4194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-exmid 4195 |
This theorem is referenced by: (None) |
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