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Theorem map0e 8438
 Description: Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 14-Jul-2022.)
Assertion
Ref Expression
map0e (𝐴𝑉 → (𝐴m ∅) = 1o)

Proof of Theorem map0e
StepHypRef Expression
1 mapdm0 8413 . 2 (𝐴𝑉 → (𝐴m ∅) = {∅})
2 df1o2 8108 . 2 1o = {∅}
31, 2syl6eqr 2872 1 (𝐴𝑉 → (𝐴m ∅) = 1o)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ∈ wcel 2107  ∅c0 4289  {csn 4559  (class class class)co 7148  1oc1o 8087   ↑m cmap 8398 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1o 8094  df-map 8400 This theorem is referenced by:  fseqenlem1  9442  infmap2  9632  pwcfsdom  9997  cfpwsdom  9998  mat0dimbas0  21067  mavmul0  21153  mavmul0g  21154  cramer0  21291  poimirlem28  34907  pwslnmlem0  39676  lincval0  44455  lco0  44467  linds0  44505
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