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Theorem 1stnpr 7449
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 7447 . 2 (1st𝐴) = dom {𝐴}
2 dmsnn0 5854 . . . . . 6 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
32biimpri 220 . . . . 5 (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 2997 . . . 4 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
54unieqd 4681 . . 3 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
6 uni0 4700 . . 3 ∅ = ∅
75, 6syl6eq 2830 . 2 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
81, 7syl5eq 2826 1 𝐴 ∈ (V × V) → (1st𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1601  wcel 2107  wne 2969  Vcvv 3398  c0 4141  {csn 4398   cuni 4671   × cxp 5353  dom cdm 5355  cfv 6135  1st c1st 7443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fv 6143  df-1st 7445
This theorem is referenced by: (None)
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