Theorem bnj521 30805

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 8505	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
2		elin 3796	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}))
3		velsn 4193	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4		eleq1 2689	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	4	biimpac 503	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐴)
6	3, 5	sylan2b 492	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝐴}) → 𝐴 ∈ 𝐴)
7	2, 6	sylbi 207	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
8	7	exlimiv 1858	. . . 4 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴 ∈ 𝐴)
9	1, 8	mto 188	. . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴})
10		n0 3931	. . 3 ⊢ ((𝐴 ∩ {𝐴}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}))
11	9, 10	mtbir 313	. 2 ⊢ ¬ (𝐴 ∩ {𝐴}) ≠ ∅
12		nne 2798	. 2 ⊢ (¬ (𝐴 ∩ {𝐴}) ≠ ∅ ↔ (𝐴 ∩ {𝐴}) = ∅)
13	11, 12	mpbi 220	1 ⊢ (𝐴 ∩ {𝐴}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794   ∩ cin 3573  ∅c0 3915  {csn 4177

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  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

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-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  bnj927  30839  bnj535  30960