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Mirrors > Home > NFE Home > Th. List > rabxm | GIF version |
Description: Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) |
Ref | Expression |
---|---|
rabxm | ⊢ A = ({x ∈ A ∣ φ} ∪ {x ∈ A ∣ ¬ φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabid2 2788 | . . 3 ⊢ (A = {x ∈ A ∣ (φ ∨ ¬ φ)} ↔ ∀x ∈ A (φ ∨ ¬ φ)) | |
2 | exmid 404 | . . . 4 ⊢ (φ ∨ ¬ φ) | |
3 | 2 | a1i 10 | . . 3 ⊢ (x ∈ A → (φ ∨ ¬ φ)) |
4 | 1, 3 | mprgbir 2684 | . 2 ⊢ A = {x ∈ A ∣ (φ ∨ ¬ φ)} |
5 | unrab 3526 | . 2 ⊢ ({x ∈ A ∣ φ} ∪ {x ∈ A ∣ ¬ φ}) = {x ∈ A ∣ (φ ∨ ¬ φ)} | |
6 | 4, 5 | eqtr4i 2376 | 1 ⊢ A = ({x ∈ A ∣ φ} ∪ {x ∈ A ∣ ¬ φ}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 357 = wceq 1642 ∈ wcel 1710 {crab 2618 ∪ cun 3207 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 |
This theorem is referenced by: (None) |
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