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Theorem bnj521 30931
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.
StepHypRef Expression
1 elirr 8543 . . . 4 ¬ 𝐴𝐴
2 elin 3829 . . . . . 6 (𝑥 ∈ (𝐴 ∩ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
3 velsn 4226 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 eleq1 2718 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
54biimpac 502 . . . . . . 7 ((𝑥𝐴𝑥 = 𝐴) → 𝐴𝐴)
63, 5sylan2b 491 . . . . . 6 ((𝑥𝐴𝑥 ∈ {𝐴}) → 𝐴𝐴)
72, 6sylbi 207 . . . . 5 (𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴𝐴)
87exlimiv 1898 . . . 4 (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴𝐴)
91, 8mto 188 . . 3 ¬ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴})
10 n0 3964 . . 3 ((𝐴 ∩ {𝐴}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}))
119, 10mtbir 312 . 2 ¬ (𝐴 ∩ {𝐴}) ≠ ∅
12 nne 2827 . 2 (¬ (𝐴 ∩ {𝐴}) ≠ ∅ ↔ (𝐴 ∩ {𝐴}) = ∅)
1311, 12mpbi 220 1 (𝐴 ∩ {𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  cin 3606  c0 3948  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by:  bnj927  30965  bnj535  31086
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