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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 8833 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = {∅}) | |
2 | df1o2 8469 | . 2 ⊢ 1o = {∅} | |
3 | 1, 2 | eqtr4di 2782 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4315 {csn 4621 (class class class)co 7402 1oc1o 8455 ↑m cmap 8817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1o 8462 df-map 8819 |
This theorem is referenced by: fseqenlem1 10016 infmap2 10210 pwcfsdom 10575 cfpwsdom 10576 mat0dimbas0 22312 mavmul0 22398 mavmul0g 22399 cramer0 22536 poimirlem28 37019 pwslnmlem0 42383 lincval0 47344 lco0 47356 linds0 47394 |
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