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Mirrors > Home > MPE Home > Th. List > map1 | Structured version Visualization version GIF version |
Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.) |
Ref | Expression |
---|---|
map1 | ⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 8424 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | oveq1i 7372 | . 2 ⊢ (1o ↑m 𝐴) = ({∅} ↑m 𝐴) |
3 | 0ex 5269 | . . 3 ⊢ ∅ ∈ V | |
4 | snmapen1 8990 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} ↑m 𝐴) ≈ 1o) | |
5 | 3, 4 | mpan 689 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↑m 𝐴) ≈ 1o) |
6 | 2, 5 | eqbrtrid 5145 | 1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3448 ∅c0 4287 {csn 4591 class class class wbr 5110 (class class class)co 7362 1oc1o 8410 ↑m cmap 8772 ≈ cen 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 |
This theorem is referenced by: (None) |
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