Theorem 1stnpr 7925

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

Assertion

Ref	Expression
1stnpr	⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅)

Proof of Theorem 1stnpr

Step	Hyp	Ref	Expression
1		1stval 7923	. 2 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}
2		dmsnn0 6154	. . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
3	2	biimpri 228	. . . . 5 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
4	3	necon1bi 2956	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
5	4	unieqd 4869	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅)
6		uni0 4884	. . 3 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2782	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅)
8	1, 7	eqtrid 2778	1 ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436  ∅c0 4280  {csn 4573  ∪ cuni 4856   × cxp 5612  dom cdm 5614  ‘cfv 6481  1^st c1st 7919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fv 6489  df-1st 7921

This theorem is referenced by: (None)