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Theorem map1 8589
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 8113 . . 3 1o = {∅}
21oveq1i 7160 . 2 (1om 𝐴) = ({∅} ↑m 𝐴)
3 0ex 5198 . . 3 ∅ ∈ V
4 snmapen1 8588 . . 3 ((∅ ∈ V ∧ 𝐴𝑉) → ({∅} ↑m 𝐴) ≈ 1o)
53, 4mpan 689 . 2 (𝐴𝑉 → ({∅} ↑m 𝐴) ≈ 1o)
62, 5eqbrtrid 5088 1 (𝐴𝑉 → (1om 𝐴) ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  Vcvv 3481  c0 4277  {csn 4551   class class class wbr 5053  (class class class)co 7150  1oc1o 8092  m cmap 8403  cen 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1o 8099  df-er 8286  df-map 8405  df-en 8507
This theorem is referenced by: (None)
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