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Theorem 2ndnpr 7927
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 7925 . 2 (2nd𝐴) = ran {𝐴}
2 rnsnn0 6161 . . . . . 6 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
32biimpri 227 . . . . 5 (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 2973 . . . 4 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
54unieqd 4880 . . 3 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
6 uni0 4897 . . 3 ∅ = ∅
75, 6eqtrdi 2793 . 2 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
81, 7eqtrid 2789 1 𝐴 ∈ (V × V) → (2nd𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  wne 2944  Vcvv 3446  c0 4283  {csn 4587   cuni 4866   × cxp 5632  ran crn 5635  cfv 6497  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-2nd 7923
This theorem is referenced by:  wlkvv  28578
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