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Theorem 2ndnpr 7690
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 7688 . 2 (2nd𝐴) = ran {𝐴}
2 rnsnn0 6053 . . . . . 6 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
32biimpri 231 . . . . 5 (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 3042 . . . 4 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
54unieqd 4839 . . 3 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
6 uni0 4853 . . 3 ∅ = ∅
75, 6syl6eq 2875 . 2 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
81, 7syl5eq 2871 1 𝐴 ∈ (V × V) → (2nd𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  wne 3014  Vcvv 3481  c0 4277  {csn 4551   cuni 4825   × cxp 5541  ran crn 5544  cfv 6344  2nd c2nd 7684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fv 6352  df-2nd 7686
This theorem is referenced by:  wlkvv  27422
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