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Theorem map1 9037
Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.)
Assertion
Ref Expression
map1 (𝐴𝑉 → (1om 𝐴) ≈ 1o)

Proof of Theorem map1
StepHypRef Expression
1 df1o2 8469 . . 3 1o = {∅}
21oveq1i 7412 . 2 (1om 𝐴) = ({∅} ↑m 𝐴)
3 0ex 5298 . . 3 ∅ ∈ V
4 snmapen1 9036 . . 3 ((∅ ∈ V ∧ 𝐴𝑉) → ({∅} ↑m 𝐴) ≈ 1o)
53, 4mpan 687 . 2 (𝐴𝑉 → ({∅} ↑m 𝐴) ≈ 1o)
62, 5eqbrtrid 5174 1 (𝐴𝑉 → (1om 𝐴) ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3466  c0 4315  {csn 4621   class class class wbr 5139  (class class class)co 7402  1oc1o 8455  m cmap 8817  cen 8933
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1o 8462  df-er 8700  df-map 8819  df-en 8937
This theorem is referenced by: (None)
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