Theorem rabnc 4343

Description: Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Assertion

Ref	Expression
rabnc	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅

Proof of Theorem rabnc

Step	Hyp	Ref	Expression
1		inrab 4263	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)}
2		pm3.24 408	. . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	2	rgenw 3083	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑)
4		rabeq0 4340	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑))
5	3, 4	mpbir 234	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅
6	1, 5	eqtri 2786	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569  ∀wral 3079  {crab 3415   ∩ cin 4091  ∅c0 4280

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3416  df-v 3455  df-dif 4049  df-in 4093  df-nul 4281

This theorem is used by:  elneldisj  4344  vtxdgoddnumeven  30146  esumrnmpt2  34753  hasheuni  34770  ddemeas  34922  ballotth  35223  jm2.22  44142