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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.) (Proof shortened by AV, 17-Jul-2022.) |
Ref | Expression |
---|---|
map1 | ⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8488 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | oveq1i 7425 | . 2 ⊢ (1o ↑m 𝐴) = ({∅} ↑m 𝐴) |
3 | 0ex 5302 | . . 3 ⊢ ∅ ∈ V | |
4 | snmapen1 9058 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} ↑m 𝐴) ≈ 1o) | |
5 | 3, 4 | mpan 689 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↑m 𝐴) ≈ 1o) |
6 | 2, 5 | eqbrtrid 5178 | 1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3470 ∅c0 4319 {csn 4625 class class class wbr 5143 (class class class)co 7415 1oc1o 8474 ↑m cmap 8839 ≈ cen 8955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 |
This theorem is referenced by: (None) |
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