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Theorem 1stnpr 7975
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 7973 . 2 (1st𝐴) = dom {𝐴}
2 dmsnn0 6199 . . . . . 6 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
32biimpri 227 . . . . 5 (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 2963 . . . 4 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
54unieqd 4915 . . 3 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
6 uni0 4932 . . 3 ∅ = ∅
75, 6eqtrdi 2782 . 2 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
81, 7eqtrid 2778 1 𝐴 ∈ (V × V) → (1st𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  wne 2934  Vcvv 3468  c0 4317  {csn 4623   cuni 4902   × cxp 5667  dom cdm 5669  cfv 6536  1st c1st 7969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-1st 7971
This theorem is referenced by: (None)
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