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Theorem 1stnpr 8018
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 8016 . 2 (1st𝐴) = dom {𝐴}
2 dmsnn0 6227 . . . . . 6 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
32biimpri 228 . . . . 5 (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 2969 . . . 4 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
54unieqd 4920 . . 3 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
6 uni0 4935 . . 3 ∅ = ∅
75, 6eqtrdi 2793 . 2 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
81, 7eqtrid 2789 1 𝐴 ∈ (V × V) → (1st𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  c0 4333  {csn 4626   cuni 4907   × cxp 5683  dom cdm 5685  cfv 6561  1st c1st 8012
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014
This theorem is referenced by: (None)
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