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Mirrors > Home > MPE Home > Th. List > rabnc | Structured version Visualization version 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 4099 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} | |
2 | pm3.24 392 | . . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
3 | 2 | rgenw 3105 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑) |
4 | rabeq0 4157 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑)) | |
5 | 3, 4 | mpbir 223 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ |
6 | 1, 5 | eqtri 2821 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 385 = wceq 1653 ∀wral 3089 {crab 3093 ∩ cin 3768 ∅c0 4115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rab 3098 df-v 3387 df-dif 3772 df-in 3776 df-nul 4116 |
This theorem is referenced by: elneldisj 4161 vtxdgoddnumeven 26803 esumrnmpt2 30646 hasheuni 30663 ddemeas 30815 ballotth 31116 jm2.22 38347 |
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