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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 3407 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} | |
2 | rabeq0 3452 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑)) | |
3 | pm3.24 693 | . . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝜑 ∧ ¬ 𝜑)) |
5 | 2, 4 | mprgbir 2535 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ |
6 | 1, 5 | eqtri 2198 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {crab 2459 ∩ cin 3128 ∅c0 3422 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rab 2464 df-v 2739 df-dif 3131 df-in 3135 df-nul 3423 |
This theorem is referenced by: (None) |
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