Theorem 1stnpr 7941

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 7939	. 2 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}
2		dmsnn0 6167	. . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
3	2	biimpri 228	. . . . 5 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
4	3	necon1bi 2961	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
5	4	unieqd 4864	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅)
6		uni0 4879	. . 3 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2788	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅)
8	1, 7	eqtrid 2784	1 ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430  ∅c0 4274  {csn 4568  ∪ cuni 4851   × cxp 5624  dom cdm 5626  ‘cfv 6494  1^st c1st 7935

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 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-iota 6450  df-fun 6496  df-fv 6502  df-1st 7937

This theorem is referenced by: (None)