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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 8469 | . . 3 ⊢ 1o = {∅} | |
2 | 1 | oveq1i 7412 | . 2 ⊢ (1o ↑m 𝐴) = ({∅} ↑m 𝐴) |
3 | 0ex 5298 | . . 3 ⊢ ∅ ∈ V | |
4 | snmapen1 9036 | . . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} ↑m 𝐴) ≈ 1o) | |
5 | 3, 4 | mpan 687 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↑m 𝐴) ≈ 1o) |
6 | 2, 5 | eqbrtrid 5174 | 1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3466 ∅c0 4315 {csn 4621 class class class wbr 5139 (class class class)co 7402 1oc1o 8455 ↑m cmap 8817 ≈ cen 8933 |
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-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 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-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 |
This theorem is referenced by: (None) |
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