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Mirrors > Home > ILE Home > Th. List > exmid0el | GIF version |
Description: Excluded middle is equivalent to decidability of ∅ being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.) |
Ref | Expression |
---|---|
exmid0el | ⊢ (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidexmid 4120 | . . 3 ⊢ (EXMID → DECID ∅ ∈ 𝑥) | |
2 | 1 | alrimiv 1846 | . 2 ⊢ (EXMID → ∀𝑥DECID ∅ ∈ 𝑥) |
3 | ax-1 6 | . . . 4 ⊢ (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) | |
4 | 3 | alimi 1431 | . . 3 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) |
5 | df-exmid 4119 | . . 3 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) | |
6 | 4, 5 | sylibr 133 | . 2 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → EXMID) |
7 | 2, 6 | impbii 125 | 1 ⊢ (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 DECID wdc 819 ∀wal 1329 ∈ wcel 1480 ⊆ wss 3071 ∅c0 3363 {csn 3527 EXMIDwem 4118 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-exmid 4119 |
This theorem is referenced by: exmidel 4128 |
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