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Theorem 1stnpr 7867
Description: Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
1stnpr 𝐴 ∈ (V × V) → (1st𝐴) = ∅)

Proof of Theorem 1stnpr
StepHypRef Expression
1 1stval 7865 . 2 (1st𝐴) = dom {𝐴}
2 dmsnn0 6125 . . . . . 6 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
32biimpri 227 . . . . 5 (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 2970 . . . 4 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
54unieqd 4858 . . 3 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
6 uni0 4875 . . 3 ∅ = ∅
75, 6eqtrdi 2792 . 2 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
81, 7eqtrid 2788 1 𝐴 ∈ (V × V) → (1st𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2104  wne 2941  Vcvv 3437  c0 4262  {csn 4565   cuni 4844   × cxp 5598  dom cdm 5600  cfv 6458  1st c1st 7861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-iota 6410  df-fun 6460  df-fv 6466  df-1st 7863
This theorem is referenced by: (None)
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