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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 4159 | . . 3 ⊢ (EXMID → DECID ∅ ∈ 𝑥) | |
2 | 1 | alrimiv 1854 | . 2 ⊢ (EXMID → ∀𝑥DECID ∅ ∈ 𝑥) |
3 | ax-1 6 | . . . 4 ⊢ (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) | |
4 | 3 | alimi 1435 | . . 3 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) |
5 | df-exmid 4158 | . . 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 820 ∀wal 1333 ∈ wcel 2128 ⊆ wss 3102 ∅c0 3395 {csn 3561 EXMIDwem 4157 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rab 2444 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-exmid 4158 |
This theorem is referenced by: exmidel 4168 |
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