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Mirrors > Home > ILE Home > Th. List > exmidexmid | GIF version |
Description: EXMID implies that an
arbitrary proposition is decidable. That is,
EXMID captures the usual meaning of excluded middle when stated in terms
of propositions.
To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 813, peircedc 884, or condc 823. (Contributed by Jim Kingdon, 18-Jun-2022.) |
Ref | Expression |
---|---|
exmidexmid | ⊢ (EXMID → DECID 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3152 | . . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} | |
2 | df-exmid 4089 | . . . 4 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥)) | |
3 | p0ex 4082 | . . . . . 6 ⊢ {∅} ∈ V | |
4 | 3 | rabex 4042 | . . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V |
5 | sseq1 3090 | . . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅})) | |
6 | eleq2 2181 | . . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})) | |
7 | 6 | dcbid 808 | . . . . . 6 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (DECID ∅ ∈ 𝑥 ↔ DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})) |
8 | 5, 7 | imbi12d 233 | . . . . 5 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))) |
9 | 4, 8 | spcv 2753 | . . . 4 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})) |
10 | 2, 9 | sylbi 120 | . . 3 ⊢ (EXMID → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})) |
11 | 1, 10 | mpi 15 | . 2 ⊢ (EXMID → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) |
12 | 0ex 4025 | . . . . 5 ⊢ ∅ ∈ V | |
13 | 12 | snid 3526 | . . . 4 ⊢ ∅ ∈ {∅} |
14 | biidd 171 | . . . . 5 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑)) | |
15 | 14 | elrab 2813 | . . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑)) |
16 | 13, 15 | mpbiran 909 | . . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
17 | 16 | dcbii 810 | . 2 ⊢ (DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ DECID 𝜑) |
18 | 11, 17 | sylib 121 | 1 ⊢ (EXMID → DECID 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 804 ∀wal 1314 = wceq 1316 ∈ wcel 1465 {crab 2397 ⊆ wss 3041 ∅c0 3333 {csn 3497 EXMIDwem 4088 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rab 2402 df-v 2662 df-dif 3043 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-exmid 4089 |
This theorem is referenced by: exmidn0m 4094 exmid0el 4097 exmidel 4098 exmidundif 4099 exmidundifim 4100 sbthlemi3 6815 sbthlemi5 6817 sbthlemi6 6818 exmidomniim 6981 exmidfodomrlemim 7025 |
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