Theorem 2ndnpr 7809

Description: Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Assertion

Ref	Expression
2ndnpr	⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅)

Proof of Theorem 2ndnpr

Step	Hyp	Ref	Expression
1		2ndval 7807	. 2 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
2		rnsnn0 6100	. . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
3	2	biimpri 227	. . . . 5 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
4	3	necon1bi 2971	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
5	4	unieqd 4850	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
6		uni0 4866	. . 3 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2795	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
8	1, 7	eqtrid 2790	1 ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  Vcvv 3422  ∅c0 4253  {csn 4558  ∪ cuni 4836   × cxp 5578  ran crn 5581  ‘cfv 6418  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-2nd 7805

This theorem is referenced by:  wlkvv  27896