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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 4316 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} | |
| 2 | pm3.24 402 | . . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
| 3 | 2 | rgenw 3065 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑) |
| 4 | rabeq0 4388 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑)) | |
| 5 | 3, 4 | mpbir 231 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ |
| 6 | 1, 5 | eqtri 2765 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∀wral 3061 {crab 3436 ∩ cin 3950 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-nul 4334 |
| This theorem is referenced by: elneldisj 4392 vtxdgoddnumeven 29571 esumrnmpt2 34069 hasheuni 34086 ddemeas 34237 ballotth 34540 jm2.22 43007 |
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