Theorem bnj521 30059

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 8388	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
2		elin 3758	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}))
3		velsn 4141	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		eleq1 2676	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	4	biimpac 502	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐴)
6	3, 5	sylan2b 491	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
7	2, 6	sylbi 206	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
8	7	exlimiv 1845	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
9	1, 8	mto 187	. . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴})
10		n0 3890	. . 3 ⊢ ((𝐴 ∩ {𝐴}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}))
11	9, 10	mtbir 312	. 2 ⊢ ¬ (𝐴 ∩ {𝐴}) ≠ ∅
12		nne 2786	. 2 ⊢ (¬ (𝐴 ∩ {𝐴}) ≠ ∅ ↔ (𝐴 ∩ {𝐴}) = ∅)
13	11, 12	mpbi 219	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780   ∩ cin 3539  ∅c0 3874  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-nul 3875  df-sn 4126  df-pr 4128

This theorem is referenced by:  bnj927  30093  bnj535  30214