Theorem rabnc 4340

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 4261	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)}
2		pm3.24 408	. . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	2	rgenw 3080	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑)
4		rabeq0 4337	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑))
5	3, 4	mpbir 234	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅
6	1, 5	eqtri 2783	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569  ∀wral 3076  {crab 3412   ∩ cin 4089  ∅c0 4278

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

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 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 4047  df-in 4091  df-nul 4279

This theorem is used by:  elneldisj  4341  vtxdgoddnumeven  30134  esumrnmpt2  34742  hasheuni  34759  ddemeas  34911  ballotth  35212  jm2.22  44044