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Theorem 2ndnpr 7064
 Description: Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
2ndnpr 𝐴 ∈ (V × V) → (2nd𝐴) = ∅)

Proof of Theorem 2ndnpr
StepHypRef Expression
1 2ndval 7062 . 2 (2nd𝐴) = ran {𝐴}
2 rnsnn0 5519 . . . . . 6 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
32biimpri 217 . . . . 5 (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 2810 . . . 4 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
54unieqd 4382 . . 3 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
6 uni0 4401 . . 3 ∅ = ∅
75, 6syl6eq 2660 . 2 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
81, 7syl5eq 2656 1 𝐴 ∈ (V × V) → (2nd𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  {csn 4125  ∪ cuni 4372   × cxp 5036  ran crn 5039  ‘cfv 5804  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-2nd 7060 This theorem is referenced by:  wlkcpr  26057  1wlkvv  40831
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