Theorem 2ndnpr 7993

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 7991	. 2 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
2		rnsnn0 6211	. . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
3	2	biimpri 231	. . . . 5 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
4	3	necon1bi 2988	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
5	4	unieqd 4887	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
6		uni0 4903	. . 3 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2816	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
8	1, 7	eqtrid 2812	1 ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457  ∅c0 4286  {csn 4591  ∪ cuni 4874   × cxp 5661  ran crn 5664  ‘cfv 6540  2^nd c2nd 7987

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-2nd 7989

This theorem is used by:  wlkvv  30010