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Theorem bnj521 32081
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 9049 . . . 4 ¬ 𝐴𝐴
2 elin 3924 . . . . . 6 (𝑥 ∈ (𝐴 ∩ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
3 velsn 4555 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 eleq1 2901 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
54biimpac 482 . . . . . . 7 ((𝑥𝐴𝑥 = 𝐴) → 𝐴𝐴)
63, 5sylan2b 596 . . . . . 6 ((𝑥𝐴𝑥 ∈ {𝐴}) → 𝐴𝐴)
72, 6sylbi 220 . . . . 5 (𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴𝐴)
87exlimiv 1931 . . . 4 (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴𝐴)
91, 8mto 200 . . 3 ¬ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴})
10 n0 4282 . . 3 ((𝐴 ∩ {𝐴}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}))
119, 10mtbir 326 . 2 ¬ (𝐴 ∩ {𝐴}) ≠ ∅
12 nne 3015 . 2 (¬ (𝐴 ∩ {𝐴}) ≠ ∅ ↔ (𝐴 ∩ {𝐴}) = ∅)
1311, 12mpbi 233 1 (𝐴 ∩ {𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1538  wex 1781  wcel 2114  wne 3011  cin 3907  c0 4265  {csn 4539
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-reg 9044
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 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-nul 4266  df-sn 4540  df-pr 4542
This theorem is referenced by:  bnj927  32114  bnj535  32236
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