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Mirrors > Home > ILE Home > Th. List > pw1dc0el | GIF version |
Description: Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.) |
Ref | Expression |
---|---|
pw1dc0el | ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 6425 | . . . . . . 7 ⊢ 1o = {∅} | |
2 | 1 | eqcomi 2181 | . . . . . 6 ⊢ {∅} = 1o |
3 | 2 | sseq2i 3182 | . . . . 5 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ⊆ 1o) |
4 | velpw 3582 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ 1o) | |
5 | 3, 4 | bitr4i 187 | . . . 4 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ∈ 𝒫 1o) |
6 | 5 | imbi1i 238 | . . 3 ⊢ ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥)) |
7 | 6 | albii 1470 | . 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥)) |
8 | df-exmid 4193 | . 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) | |
9 | df-ral 2460 | . 2 ⊢ (∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥)) | |
10 | 7, 8, 9 | 3bitr4i 212 | 1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 DECID wdc 834 ∀wal 1351 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 ∅c0 3422 𝒫 cpw 3575 {csn 3592 EXMIDwem 4192 1oc1o 6405 |
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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-exmid 4193 df-suc 4369 df-1o 6412 |
This theorem is referenced by: pw1dc1 6908 |
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