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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 4047 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} | |
2 | pm3.24 389 | . . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
3 | 2 | rgenw 3073 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑) |
4 | rabeq0 4103 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑)) | |
5 | 3, 4 | mpbir 221 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ |
6 | 1, 5 | eqtri 2793 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 382 = wceq 1631 ∀wral 3061 {crab 3065 ∩ cin 3722 ∅c0 4063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rab 3070 df-v 3353 df-dif 3726 df-in 3730 df-nul 4064 |
This theorem is referenced by: elneldisj 4107 elneldisjOLD 4109 vtxdgoddnumeven 26684 esumrnmpt2 30470 hasheuni 30487 ddemeas 30639 ballotth 30939 jm2.22 38088 |
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