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Theorem map1 9059
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 8488 . . 3 1o = {∅}
21oveq1i 7425 . 2 (1om 𝐴) = ({∅} ↑m 𝐴)
3 0ex 5302 . . 3 ∅ ∈ V
4 snmapen1 9058 . . 3 ((∅ ∈ V ∧ 𝐴𝑉) → ({∅} ↑m 𝐴) ≈ 1o)
53, 4mpan 689 . 2 (𝐴𝑉 → ({∅} ↑m 𝐴) ≈ 1o)
62, 5eqbrtrid 5178 1 (𝐴𝑉 → (1om 𝐴) ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3470  c0 4319  {csn 4625   class class class wbr 5143  (class class class)co 7415  1oc1o 8474  m cmap 8839  cen 8955
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 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
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 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1o 8481  df-er 8719  df-map 8841  df-en 8959
This theorem is referenced by: (None)
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