![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > map1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
map1 | ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ≈ 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovexd 6844 | . 2 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ∈ V) | |
2 | df1o2 7743 | . . . 4 ⊢ 1𝑜 = {∅} | |
3 | p0ex 5002 | . . . 4 ⊢ {∅} ∈ V | |
4 | 2, 3 | eqeltri 2835 | . . 3 ⊢ 1𝑜 ∈ V |
5 | 4 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → 1𝑜 ∈ V) |
6 | 0ex 4942 | . . 3 ⊢ ∅ ∈ V | |
7 | 6 | 2a1i 12 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) → ∅ ∈ V)) |
8 | xpexg 7126 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V) | |
9 | 3, 8 | mpan2 709 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V) |
10 | 9 | a1d 25 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V)) |
11 | el1o 7750 | . . . . 5 ⊢ (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅) | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅)) |
13 | 2 | oveq1i 6824 | . . . . . . 7 ⊢ (1𝑜 ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴) |
14 | 13 | eleq2i 2831 | . . . . . 6 ⊢ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴)) |
15 | elmapg 8038 | . . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) | |
16 | 3, 15 | mpan 708 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) |
17 | 14, 16 | syl5bb 272 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) |
18 | 6 | fconst2 6635 | . . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅})) |
19 | 17, 18 | syl6rbb 277 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴))) |
20 | 12, 19 | anbi12d 749 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)))) |
21 | ancom 465 | . . 3 ⊢ ((𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)) ↔ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ∧ 𝑦 = ∅)) | |
22 | 20, 21 | syl6rbb 277 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅})))) |
23 | 1, 5, 7, 10, 22 | en2d 8159 | 1 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 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) |
Copyright terms: Public domain | W3C validator |