Theorem map1 8306

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 7840	. . 3 ⊢ 1o = {∅}
2	1	oveq1i 6916	. 2 ⊢ (1o ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
3		0ex 5015	. . 3 ⊢ ∅ ∈ V
4		snmapen1 8305	. . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} ↑𝑚 𝐴) ≈ 1o)
5	3, 4	mpan 683	. 2 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↑𝑚 𝐴) ≈ 1o)
6	2, 5	syl5eqbr 4909	1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o)

Syntax hints:   → wi 4   ∈ wcel 2166  Vcvv 3415  ∅c0 4145  {csn 4398   class class class wbr 4874  (class class class)co 6906  1_oc1o 7820   ↑_𝑚 cmap 8123   ≈ cen 8220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1o 7827  df-er 8010  df-map 8125  df-en 8224

This theorem is referenced by: (None)