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Mirrors > Home > MPE Home > Th. List > ficard | Structured version Visualization version GIF version |
Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ficard | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 7865 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | carden 9252 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴 ≈ 𝑥)) | |
3 | cardnn 8672 | . . . . . . . 8 ⊢ (𝑥 ∈ ω → (card‘𝑥) = 𝑥) | |
4 | eqtr 2629 | . . . . . . . . 9 ⊢ (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥) | |
5 | 4 | expcom 450 | . . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) = 𝑥)) |
6 | 3, 5 | syl 17 | . . . . . . 7 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) = 𝑥)) |
7 | eleq1a 2683 | . . . . . . 7 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω)) | |
8 | 6, 7 | syld 46 | . . . . . 6 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω)) |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω)) |
10 | 2, 9 | sylbird 249 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω)) |
11 | 10 | rexlimdva 3013 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω)) |
12 | 1, 11 | syl5bi 231 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)) |
13 | cardnn 8672 | . . . . . . . 8 ⊢ ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴)) | |
14 | 13 | eqcomd 2616 | . . . . . . 7 ⊢ ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴))) |
15 | 14 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴))) |
16 | carden 9252 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴))) | |
17 | 15, 16 | mpbid 221 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴)) |
18 | 17 | ex 449 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴))) |
19 | 18 | ancld 574 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)))) |
20 | breq2 4587 | . . . . 5 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ (card‘𝐴))) | |
21 | 20 | rspcev 3282 | . . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
22 | 21, 1 | sylibr 223 | . . 3 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → 𝐴 ∈ Fin) |
23 | 19, 22 | syl6 34 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ∈ Fin)) |
24 | 12, 23 | impbid 201 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 ωcom 6957 ≈ cen 7838 Fincfn 7841 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-ac2 9168 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-om 6958 df-wrecs 7294 df-recs 7355 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-ac 8822 |
This theorem is referenced by: cfpwsdom 9285 |
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