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Theorem map1 8203
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.)
Assertion
Ref Expression
map1 (𝐴𝑉 → (1𝑜𝑚 𝐴) ≈ 1𝑜)

Proof of Theorem map1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 6844 . 2 (𝐴𝑉 → (1𝑜𝑚 𝐴) ∈ V)
2 df1o2 7743 . . . 4 1𝑜 = {∅}
3 p0ex 5002 . . . 4 {∅} ∈ V
42, 3eqeltri 2835 . . 3 1𝑜 ∈ V
54a1i 11 . 2 (𝐴𝑉 → 1𝑜 ∈ V)
6 0ex 4942 . . 3 ∅ ∈ V
762a1i 12 . 2 (𝐴𝑉 → (𝑥 ∈ (1𝑜𝑚 𝐴) → ∅ ∈ V))
8 xpexg 7126 . . . 4 ((𝐴𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
93, 8mpan2 709 . . 3 (𝐴𝑉 → (𝐴 × {∅}) ∈ V)
109a1d 25 . 2 (𝐴𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V))
11 el1o 7750 . . . . 5 (𝑦 ∈ 1𝑜𝑦 = ∅)
1211a1i 11 . . . 4 (𝐴𝑉 → (𝑦 ∈ 1𝑜𝑦 = ∅))
132oveq1i 6824 . . . . . . 7 (1𝑜𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
1413eleq2i 2831 . . . . . 6 (𝑥 ∈ (1𝑜𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
15 elmapg 8038 . . . . . . 7 (({∅} ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
163, 15mpan 708 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
1714, 16syl5bb 272 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (1𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
186fconst2 6635 . . . . 5 (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
1917, 18syl6rbb 277 . . . 4 (𝐴𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1𝑜𝑚 𝐴)))
2012, 19anbi12d 749 . . 3 (𝐴𝑉 → ((𝑦 ∈ 1𝑜𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜𝑚 𝐴))))
21 ancom 465 . . 3 ((𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜𝑚 𝐴)) ↔ (𝑥 ∈ (1𝑜𝑚 𝐴) ∧ 𝑦 = ∅))
2220, 21syl6rbb 277 . 2 (𝐴𝑉 → ((𝑥 ∈ (1𝑜𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1𝑜𝑥 = (𝐴 × {∅}))))
231, 5, 7, 10, 22en2d 8159 1 (𝐴𝑉 → (1𝑜𝑚 𝐴) ≈ 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  {csn 4321   class class class wbr 4804   × cxp 5264  wf 6045  (class class class)co 6814  1𝑜c1o 7723  𝑚 cmap 8025  cen 8120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1o 7730  df-map 8027  df-en 8124
This theorem is referenced by: (None)
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