Theorem bnj521 32616

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj521	⊢ (𝐴 ∩ {𝐴}) = ∅

Proof of Theorem bnj521

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elirr 9286	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
2		elin 3899	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}))
3		velsn 4574	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		eleq1 2826	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	4	biimpac 478	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐴)
6	3, 5	sylan2b 593	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
7	2, 6	sylbi 216	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
8	7	exlimiv 1934	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
9	1, 8	mto 196	. . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴})
10		n0 4277	. . 3 ⊢ ((𝐴 ∩ {𝐴}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}))
11	9, 10	mtbir 322	. 2 ⊢ ¬ (𝐴 ∩ {𝐴}) ≠ ∅
12		nne 2946	. 2 ⊢ (¬ (𝐴 ∩ {𝐴}) ≠ ∅ ↔ (𝐴 ∩ {𝐴}) = ∅)
13	11, 12	mpbi 229	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942   ∩ cin 3882  ∅c0 4253  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-reg 9281

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-sn 4559  df-pr 4561

This theorem is referenced by:  bnj927  32649  bnj535  32770