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Mirrors > Home > MPE Home > Th. List > isnum3 | Structured version Visualization version GIF version |
Description: A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
isnum3 | ⊢ (𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardid2 9992 | . 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
2 | cardon 9983 | . . 3 ⊢ (card‘𝐴) ∈ On | |
3 | isnumi 9985 | . . 3 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴) → 𝐴 ∈ dom card) | |
4 | 2, 3 | mpan 688 | . 2 ⊢ ((card‘𝐴) ≈ 𝐴 → 𝐴 ∈ dom card) |
5 | 1, 4 | impbii 208 | 1 ⊢ (𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 class class class wbr 5152 dom cdm 5681 Oncon0 6375 ‘cfv 6553 ≈ cen 8970 cardccrd 9974 |
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 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-ord 6378 df-on 6379 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-en 8974 df-card 9978 |
This theorem is referenced by: ttukey2g 10555 |
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