![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 837 | . . . . . 6 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
2 | 1 | a1d 22 | . . . . 5 ⊢ (DECID 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))) |
3 | 2 | alimi 1466 | . . . 4 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))) |
4 | df-ral 2473 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))) | |
5 | 3, 4 | sylibr 134 | . . 3 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑)) |
6 | rabid2 2667 | . . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑)) | |
7 | 5, 6 | sylibr 134 | . 2 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}) |
8 | unrab 3421 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} | |
9 | 7, 8 | eqtr4di 2240 | 1 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 709 DECID wdc 835 ∀wal 1362 = wceq 1364 ∈ wcel 2160 ∀wral 2468 {crab 2472 ∪ cun 3142 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rab 2477 df-v 2754 df-un 3148 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |