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Theorem snmapen1 8587
Description: Set exponentiation: a singleton to any set is equinumerous to ordinal 1. (Proposed by BJ, 17-Jul-2022.) (Contributed by AV, 17-Jul-2022.)
Assertion
Ref Expression
snmapen1 ((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ≈ 1o)

Proof of Theorem snmapen1
StepHypRef Expression
1 snmapen 8586 . 2 ((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})
2 ensn1g 8570 . . 3 (𝐴𝑉 → {𝐴} ≈ 1o)
32adantr 484 . 2 ((𝐴𝑉𝐵𝑊) → {𝐴} ≈ 1o)
4 entr 8557 . 2 ((({𝐴} ↑m 𝐵) ≈ {𝐴} ∧ {𝐴} ≈ 1o) → ({𝐴} ↑m 𝐵) ≈ 1o)
51, 3, 4syl2anc 587 1 ((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  {csn 4550   class class class wbr 5052  (class class class)co 7149  1oc1o 8091  m cmap 8402  cen 8502
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 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1o 8098  df-er 8285  df-map 8404  df-en 8506
This theorem is referenced by:  map1  8588
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