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Theorem map0e 8051
 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 8028 . 2 (𝐴𝑉 → (𝐴𝑚 ∅) = {∅})
2 df1o2 7730 . 2 1𝑜 = {∅}
31, 2syl6eqr 2823 1 (𝐴𝑉 → (𝐴𝑚 ∅) = 1𝑜)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ∅c0 4063  {csn 4317  (class class class)co 6796  1𝑜c1o 7710   ↑𝑚 cmap 8013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1o 7717  df-map 8015 This theorem is referenced by:  fseqenlem1  9051  infmap2  9246  pwcfsdom  9611  cfpwsdom  9612  mat0dimbas0  20490  mavmul0  20576  mavmul0g  20577  cramer0  20716  poimirlem28  33770  pwslnmlem0  38187  lincval0  42729  lco0  42741  linds0  42779
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