ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1st0 GIF version

Theorem 1st0 6138
Description: The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
Assertion
Ref Expression
1st0 (1st ‘∅) = ∅

Proof of Theorem 1st0
StepHypRef Expression
1 0ex 4127 . . 3 ∅ ∈ V
2 1stvalg 6136 . . 3 (∅ ∈ V → (1st ‘∅) = dom {∅})
31, 2ax-mp 5 . 2 (1st ‘∅) = dom {∅}
4 dmsn0 5091 . . 3 dom {∅} = ∅
54unieqi 3817 . 2 dom {∅} =
6 uni0 3834 . 2 ∅ = ∅
73, 5, 63eqtri 2202 1 (1st ‘∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  Vcvv 2737  c0 3422  {csn 3591   cuni 3807  dom cdm 4622  cfv 5211  1st c1st 6132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fv 5219  df-1st 6134
This theorem is referenced by:  0npr  7460
  Copyright terms: Public domain W3C validator