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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 8110 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | oveq1i 7160 | . 2 ⊢ (1o ↑m 𝐴) = ({∅} ↑m 𝐴) |
3 | 0ex 5203 | . . 3 ⊢ ∅ ∈ V | |
4 | snmapen1 8585 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} ↑m 𝐴) ≈ 1o) | |
5 | 3, 4 | mpan 688 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↑m 𝐴) ≈ 1o) |
6 | 2, 5 | eqbrtrid 5093 | 1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 {csn 4560 class class class wbr 5058 (class class class)co 7150 1oc1o 8089 ↑m cmap 8400 ≈ cen 8500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1o 8096 df-er 8283 df-map 8402 df-en 8504 |
This theorem is referenced by: (None) |
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