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Theorem bnj521 29865
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 8365 . . . 4 ¬ 𝐴𝐴
2 elin 3757 . . . . . 6 (𝑥 ∈ (𝐴 ∩ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
3 velsn 4140 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
4 eleq1 2675 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
54biimpac 501 . . . . . . 7 ((𝑥𝐴𝑥 = 𝐴) → 𝐴𝐴)
63, 5sylan2b 490 . . . . . 6 ((𝑥𝐴𝑥 ∈ {𝐴}) → 𝐴𝐴)
72, 6sylbi 205 . . . . 5 (𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴𝐴)
87exlimiv 1844 . . . 4 (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}) → 𝐴𝐴)
91, 8mto 186 . . 3 ¬ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴})
10 n0 3889 . . 3 ((𝐴 ∩ {𝐴}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝐴}))
119, 10mtbir 311 . 2 ¬ (𝐴 ∩ {𝐴}) ≠ ∅
12 nne 2785 . 2 (¬ (𝐴 ∩ {𝐴}) ≠ ∅ ↔ (𝐴 ∩ {𝐴}) = ∅)
1311, 12mpbi 218 1 (𝐴 ∩ {𝐴}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1474  wex 1694  wcel 1976  wne 2779  cin 3538  c0 3873  {csn 4124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-reg 8357
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-nul 3874  df-sn 4125  df-pr 4127
This theorem is referenced by:  bnj927  29899  bnj535  30020
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