Theorem bnj521 32009

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 9063	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
2		elin 4171	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}))
3		velsn 4585	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		eleq1 2902	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	4	biimpac 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐴)
6	3, 5	sylan2b 595	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
7	2, 6	sylbi 219	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
8	7	exlimiv 1931	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
9	1, 8	mto 199	. . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴})
10		n0 4312	. . 3 ⊢ ((𝐴 ∩ {𝐴}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}))
11	9, 10	mtbir 325	. 2 ⊢ ¬ (𝐴 ∩ {𝐴}) ≠ ∅
12		nne 3022	. 2 ⊢ (¬ (𝐴 ∩ {𝐴}) ≠ ∅ ↔ (𝐴 ∩ {𝐴}) = ∅)
13	11, 12	mpbi 232	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018   ∩ cin 3937  ∅c0 4293  {csn 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-reg 9058

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-nul 4294  df-sn 4570  df-pr 4572

This theorem is referenced by:  bnj927  32042  bnj535  32164