Theorem map1 8830

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	⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o)

Proof of Theorem map1

Step	Hyp	Ref	Expression
1		df1o2 8304	. . 3 ⊢ 1o = {∅}
2	1	oveq1i 7285	. 2 ⊢ (1o ↑m 𝐴) = ({∅} ↑m 𝐴)
3		0ex 5231	. . 3 ⊢ ∅ ∈ V
4		snmapen1 8829	. . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} ↑m 𝐴) ≈ 1o)
5	3, 4	mpan 687	. 2 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↑m 𝐴) ≈ 1o)
6	2, 5	eqbrtrid 5109	1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  {csn 4561   class class class wbr 5074  (class class class)co 7275  1_oc1o 8290   ↑_m cmap 8615   ≈ cen 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-er 8498  df-map 8617  df-en 8734

This theorem is referenced by: (None)