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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 | ovex 6577 | . . 3 ⊢ (1𝑜 ↑𝑚 𝐴) ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ∈ V) |
3 | df1o2 7459 | . . . 4 ⊢ 1𝑜 = {∅} | |
4 | p0ex 4779 | . . . 4 ⊢ {∅} ∈ V | |
5 | 3, 4 | eqeltri 2684 | . . 3 ⊢ 1𝑜 ∈ V |
6 | 5 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → 1𝑜 ∈ V) |
7 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
8 | 7 | 2a1i 12 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) → ∅ ∈ V)) |
9 | xpexg 6858 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V) | |
10 | 4, 9 | mpan2 703 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V) |
11 | 10 | a1d 25 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V)) |
12 | el1o 7466 | . . . . 5 ⊢ (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅) | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅)) |
14 | 3 | oveq1i 6559 | . . . . . . 7 ⊢ (1𝑜 ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴) |
15 | 14 | eleq2i 2680 | . . . . . 6 ⊢ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴)) |
16 | elmapg 7757 | . . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) | |
17 | 4, 16 | mpan 702 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) |
18 | 15, 17 | syl5bb 271 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) |
19 | 7 | fconst2 6375 | . . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅})) |
20 | 18, 19 | syl6rbb 276 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴))) |
21 | 13, 20 | anbi12d 743 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)))) |
22 | ancom 465 | . . 3 ⊢ ((𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)) ↔ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ∧ 𝑦 = ∅)) | |
23 | 21, 22 | syl6rbb 276 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅})))) |
24 | 2, 6, 8, 11, 23 | en2d 7877 | 1 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ≈ 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 class class class wbr 4583 × cxp 5036 ⟶wf 5800 (class class class)co 6549 1𝑜c1o 7440 ↑𝑚 cmap 7744 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1o 7447 df-map 7746 df-en 7842 |
This theorem is referenced by: (None) |
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