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Theorem 1stnpr 7124
 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 7122 . 2 (1st𝐴) = dom {𝐴}
2 dmsnn0 5564 . . . . . 6 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
32biimpri 218 . . . . 5 (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V))
43necon1bi 2818 . . . 4 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
54unieqd 4417 . . 3 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
6 uni0 4436 . . 3 ∅ = ∅
75, 6syl6eq 2671 . 2 𝐴 ∈ (V × V) → dom {𝐴} = ∅)
81, 7syl5eq 2667 1 𝐴 ∈ (V × V) → (1st𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  Vcvv 3189  ∅c0 3896  {csn 4153  ∪ cuni 4407   × cxp 5077  dom cdm 5079  ‘cfv 5852  1st c1st 7118 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fv 5860  df-1st 7120 This theorem is referenced by: (None)
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