Theorem 2ndnpr 8035

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 8033	. 2 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
2		rnsnn0 6239	. . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
3	2	biimpri 228	. . . . 5 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
4	3	necon1bi 2975	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
5	4	unieqd 4944	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
6		uni0 4959	. . 3 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2796	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
8	1, 7	eqtrid 2792	1 ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488  ∅c0 4352  {csn 4648  ∪ cuni 4931   × cxp 5698  ran crn 5701  ‘cfv 6573  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-2nd 8031

This theorem is referenced by:  wlkvv  29663