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Mirrors > Home > ILE Home > Th. List > rabnc | GIF version |
Description: Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) |
Ref | Expression |
---|---|
rabnc | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inrab 3389 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} | |
2 | rabeq0 3433 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑)) | |
3 | pm3.24 683 | . . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝜑 ∧ ¬ 𝜑)) |
5 | 2, 4 | mprgbir 2522 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ |
6 | 1, 5 | eqtri 2185 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1342 ∈ wcel 2135 {crab 2446 ∩ cin 3110 ∅c0 3404 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rab 2451 df-v 2723 df-dif 3113 df-in 3117 df-nul 3405 |
This theorem is referenced by: (None) |
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