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Theorem map1 8577
 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 8101 . . 3 1o = {∅}
21oveq1i 7145 . 2 (1om 𝐴) = ({∅} ↑m 𝐴)
3 0ex 5175 . . 3 ∅ ∈ V
4 snmapen1 8576 . . 3 ((∅ ∈ V ∧ 𝐴𝑉) → ({∅} ↑m 𝐴) ≈ 1o)
53, 4mpan 689 . 2 (𝐴𝑉 → ({∅} ↑m 𝐴) ≈ 1o)
62, 5eqbrtrid 5065 1 (𝐴𝑉 → (1om 𝐴) ≈ 1o)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  {csn 4525   class class class wbr 5030  (class class class)co 7135  1oc1o 8080   ↑m cmap 8391   ≈ cen 8491 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1o 8087  df-er 8274  df-map 8393  df-en 8495 This theorem is referenced by: (None)
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