Theorem rabnc 4106

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}) = ∅

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabnc

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}) = ∅

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