Theorem 2ndnpr 8004

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 8002	. 2 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
2		rnsnn0 6208	. . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
3	2	biimpri 231	. . . . 5 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
4	3	necon1bi 2984	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅)
5	4	unieqd 4880	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅)
6		uni0 4896	. . 3 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2812	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅)
8	1, 7	eqtrid 2808	1 ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  Vcvv 3451  ∅c0 4279  {csn 4584  ∪ cuni 4867   × cxp 5649  ran crn 5652  ‘cfv 6537  2^nd c2nd 7998

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fv 6545  df-2nd 8000

This theorem is used by:  wlkvv  30200