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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 8860 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = {∅}) | |
2 | df1o2 8493 | . 2 ⊢ 1o = {∅} | |
3 | 1, 2 | eqtr4di 2786 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∅c0 4323 {csn 4629 (class class class)co 7420 1oc1o 8479 ↑m cmap 8844 |
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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1o 8486 df-map 8846 |
This theorem is referenced by: fseqenlem1 10047 infmap2 10241 pwcfsdom 10606 cfpwsdom 10607 mat0dimbas0 22367 mavmul0 22453 mavmul0g 22454 cramer0 22591 poimirlem28 37121 pwslnmlem0 42515 lincval0 47483 lco0 47495 linds0 47533 |
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