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Theorem bj-intexr 12940
Description: intexr 4043 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-intexr ( 𝐴 ∈ V → 𝐴 ≠ ∅)

Proof of Theorem bj-intexr
StepHypRef Expression
1 bj-vprc 12928 . . 3 ¬ V ∈ V
2 inteq 3742 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 3753 . . . . 5 ∅ = V
42, 3syl6eq 2164 . . . 4 (𝐴 = ∅ → 𝐴 = V)
54eleq1d 2184 . . 3 (𝐴 = ∅ → ( 𝐴 ∈ V ↔ V ∈ V))
61, 5mtbiri 647 . 2 (𝐴 = ∅ → ¬ 𝐴 ∈ V)
76necon2ai 2337 1 ( 𝐴 ∈ V → 𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  wne 2283  Vcvv 2658  c0 3331   cint 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bdn 12849  ax-bdel 12853  ax-bdsep 12916
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-v 2660  df-dif 3041  df-nul 3332  df-int 3740
This theorem is referenced by: (None)
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