Theorem map0e 8160

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	⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1o)

Proof of Theorem map0e

Step	Hyp	Ref	Expression
1		mapdm0 8137	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅})
2		df1o2 7839	. 2 ⊢ 1o = {∅}
3	1, 2	syl6eqr 2879	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1o)

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  ∅c0 4144  {csn 4397  (class class class)co 6905  1_oc1o 7819   ↑_𝑚 cmap 8122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1o 7826  df-map 8124

This theorem is referenced by:  fseqenlem1  9160  infmap2  9355  pwcfsdom  9720  cfpwsdom  9721  mat0dimbas0  20640  mavmul0  20726  mavmul0g  20727  cramer0  20866  poimirlem28  33981  pwslnmlem0  38504  lincval0  43051  lco0  43063  linds0  43101