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Theorem 2ndnpr 7319
 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 7317 . 2 (2nd𝐴) = ran {𝐴}
2 rnsnn0 5742 . . . . . 6 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
32biimpri 218 . . . . 5 (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 2970 . . . 4 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
54unieqd 4582 . . 3 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
6 uni0 4599 . . 3 ∅ = ∅
75, 6syl6eq 2820 . 2 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
81, 7syl5eq 2816 1 𝐴 ∈ (V × V) → (2nd𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1630   ∈ wcel 2144   ≠ wne 2942  Vcvv 3349  ∅c0 4061  {csn 4314  ∪ cuni 4572   × cxp 5247  ran crn 5250  ‘cfv 6031  2nd c2nd 7313 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-2nd 7315 This theorem is referenced by:  wlkvv  26756
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