Theorem 1stnpr 8026

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 8024	. 2 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}
2		dmsnn0 6220	. . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
3	2	biimpri 231	. . . . 5 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
4	3	necon1bi 2987	. . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
5	4	unieqd 4883	. . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅)
6		uni0 4899	. . 3 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2814	. 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅)
8	1, 7	eqtrid 2810	1 ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  {csn 4588  ∪ cuni 4870   × cxp 5653  dom cdm 5655  ‘cfv 6555  1^st c1st 8020

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fv 6563  df-1st 8022

This theorem is used by: (None)