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Mirrors > Home > ILE Home > Th. List > rabxmdc | GIF version |
Description: Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.) |
Ref | Expression |
---|---|
rabxmdc | ⊢ (∀𝑥DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmiddc 821 | . . . . . 6 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
2 | 1 | a1d 22 | . . . . 5 ⊢ (DECID 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))) |
3 | 2 | alimi 1431 | . . . 4 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))) |
4 | df-ral 2421 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))) | |
5 | 3, 4 | sylibr 133 | . . 3 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑)) |
6 | rabid2 2607 | . . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑)) | |
7 | 5, 6 | sylibr 133 | . 2 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}) |
8 | unrab 3347 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} | |
9 | 7, 8 | syl6eqr 2190 | 1 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 697 DECID wdc 819 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∀wral 2416 {crab 2420 ∪ cun 3069 |
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-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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
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-ral 2421 df-rab 2425 df-v 2688 df-un 3075 |
This theorem is referenced by: (None) |
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