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Mirrors > Home > ILE Home > Th. List > map0e | 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.) |
Ref | Expression |
---|---|
map0e | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4055 | . . . 4 ⊢ ∅ ∈ V | |
2 | elmapg 6555 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴)) | |
3 | 1, 2 | mpan2 421 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴)) |
4 | f0bi 5315 | . . . 4 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 = ∅) | |
5 | el1o 6334 | . . . 4 ⊢ (𝑓 ∈ 1o ↔ 𝑓 = ∅) | |
6 | 4, 5 | bitr4i 186 | . . 3 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 ∈ 1o) |
7 | 3, 6 | syl6bb 195 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓 ∈ 1o)) |
8 | 7 | eqrdv 2137 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∅c0 3363 ⟶wf 5119 (class class class)co 5774 1oc1o 6306 ↑𝑚 cmap 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1o 6313 df-map 6544 |
This theorem is referenced by: (None) |
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