Theorem rabnc 3962

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 3899	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)}
2		pm3.24 926	. . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	2	rgenw 2924	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑)
4		rabeq0 3957	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜑))
5	3, 4	mpbir 221	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅
6	1, 5	eqtri 2644	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483  ∀wral 2912  {crab 2916   ∩ cin 3573  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916

This theorem is referenced by:  elneldisj  3963  elneldisjOLD  3965  vtxdgoddnumeven  26449  esumrnmpt2  30130  hasheuni  30147  ddemeas  30299  ballotth  30599  jm2.22  37562