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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 8404 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = {∅}) | |
2 | df1o2 8099 | . 2 ⊢ 1o = {∅} | |
3 | 1, 2 | eqtr4di 2851 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∅c0 4243 {csn 4525 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1o 8085 df-map 8391 |
This theorem is referenced by: fseqenlem1 9435 infmap2 9629 pwcfsdom 9994 cfpwsdom 9995 mat0dimbas0 21071 mavmul0 21157 mavmul0g 21158 cramer0 21295 poimirlem28 35085 pwslnmlem0 40035 lincval0 44824 lco0 44836 linds0 44874 |
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