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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 4128 | . . 3 ⊢ (EXMID → DECID ∅ ∈ 𝑥) | |
2 | 1 | alrimiv 1847 | . 2 ⊢ (EXMID → ∀𝑥DECID ∅ ∈ 𝑥) |
3 | ax-1 6 | . . . 4 ⊢ (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) | |
4 | 3 | alimi 1432 | . . 3 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) |
5 | df-exmid 4127 | . . 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 1330 ∈ wcel 1481 ⊆ wss 3076 ∅c0 3368 {csn 3532 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: exmidel 4136 |
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