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Mirrors > Home > MPE Home > Th. List > carden2 | Structured version Visualization version GIF version |
Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10575, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.) |
Ref | Expression |
---|---|
carden2 | ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carddom2 10001 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵)) | |
2 | carddom2 10001 | . . . 4 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ 𝐵 ≼ 𝐴)) | |
3 | 2 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ 𝐵 ≼ 𝐴)) |
4 | 1, 3 | anbi12d 631 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)) ↔ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴))) |
5 | eqss 3995 | . . 3 ⊢ ((card‘𝐴) = (card‘𝐵) ↔ ((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴))) | |
6 | 5 | bicomi 223 | . 2 ⊢ (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)) ↔ (card‘𝐴) = (card‘𝐵)) |
7 | sbthb 9119 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ 𝐴 ≈ 𝐵) | |
8 | 4, 6, 7 | 3bitr3g 313 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 class class class wbr 5148 dom cdm 5678 ‘cfv 6548 ≈ cen 8961 ≼ cdom 8962 cardccrd 9959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-card 9963 |
This theorem is referenced by: cardsdom2 10012 pm54.43lem 10024 sdom2en01 10326 fin23lem22 10351 fin1a2lem9 10432 pwfseqlem4 10686 hashen 14339 |
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