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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 8842 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = {∅}) | |
2 | df1o2 8479 | . 2 ⊢ 1o = {∅} | |
3 | 1, 2 | eqtr4di 2789 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∅c0 4322 {csn 4628 (class class class)co 7412 1oc1o 8465 ↑m cmap 8826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1o 8472 df-map 8828 |
This theorem is referenced by: fseqenlem1 10025 infmap2 10219 pwcfsdom 10584 cfpwsdom 10585 mat0dimbas0 22288 mavmul0 22374 mavmul0g 22375 cramer0 22512 poimirlem28 36980 pwslnmlem0 42296 lincval0 47258 lco0 47270 linds0 47308 |
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