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Mirrors > Home > MPE Home > Th. List > map0e | Structured version Visualization version GIF version |
Description: Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 14-Jul-2022.) |
Ref | Expression |
---|---|
map0e | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdm0 8410 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = {∅}) | |
2 | df1o2 8105 | . 2 ⊢ 1o = {∅} | |
3 | 1, 2 | syl6eqr 2871 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∅c0 4288 {csn 4557 (class class class)co 7145 1oc1o 8084 ↑m cmap 8395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1o 8091 df-map 8397 |
This theorem is referenced by: fseqenlem1 9438 infmap2 9628 pwcfsdom 9993 cfpwsdom 9994 mat0dimbas0 21003 mavmul0 21089 mavmul0g 21090 cramer0 21227 poimirlem28 34801 pwslnmlem0 39569 lincval0 44398 lco0 44410 linds0 44448 |
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