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Theorem map1 9065
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 8494 . . 3 1o = {∅}
21oveq1i 7430 . 2 (1om 𝐴) = ({∅} ↑m 𝐴)
3 0ex 5307 . . 3 ∅ ∈ V
4 snmapen1 9064 . . 3 ((∅ ∈ V ∧ 𝐴𝑉) → ({∅} ↑m 𝐴) ≈ 1o)
53, 4mpan 689 . 2 (𝐴𝑉 → ({∅} ↑m 𝐴) ≈ 1o)
62, 5eqbrtrid 5183 1 (𝐴𝑉 → (1om 𝐴) ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3471  c0 4323  {csn 4629   class class class wbr 5148  (class class class)co 7420  1oc1o 8480  m cmap 8845  cen 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8487  df-er 8725  df-map 8847  df-en 8965
This theorem is referenced by: (None)
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