Theorem 2ndnpr 7947

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 7945	. 2 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
2		rnsnn0 6172	. . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
3	2	biimpri 228	. . . . 5 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
4	3	necon1bi 2960	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
5	4	unieqd 4863	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
6		uni0 4878	. . 3 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2787	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
8	1, 7	eqtrid 2783	1 ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429  ∅c0 4273  {csn 4567  ∪ cuni 4850   × cxp 5629  ran crn 5632  ‘cfv 6498  2^nd c2nd 7941

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-2nd 7943

This theorem is referenced by:  wlkvv  29695