Theorem bnj521 32117

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 9045	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
2		elin 3897	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}))
3		velsn 4541	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		eleq1 2877	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	4	biimpac 482	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐴)
6	3, 5	sylan2b 596	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
7	2, 6	sylbi 220	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
8	7	exlimiv 1931	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
9	1, 8	mto 200	. . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴})
10		n0 4260	. . 3 ⊢ ((𝐴 ∩ {𝐴}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}))
11	9, 10	mtbir 326	. 2 ⊢ ¬ (𝐴 ∩ {𝐴}) ≠ ∅
12		nne 2991	. 2 ⊢ (¬ (𝐴 ∩ {𝐴}) ≠ ∅ ↔ (𝐴 ∩ {𝐴}) = ∅)
13	11, 12	mpbi 233	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987   ∩ cin 3880  ∅c0 4243  {csn 4525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-reg 9040

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-nul 4244  df-sn 4526  df-pr 4528

This theorem is referenced by:  bnj927  32150  bnj535  32272