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Mirrors > Home > ILE Home > Th. List > isfi | GIF version |
Description: Express "𝐴 is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose "Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.) |
Ref | Expression |
---|---|
isfi | ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fin 6630 | . . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} | |
2 | 1 | eleq2i 2204 | . 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}) |
3 | relen 6631 | . . . . 5 ⊢ Rel ≈ | |
4 | 3 | brrelex1i 4577 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
5 | 4 | rexlimivw 2543 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
6 | breq1 3927 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥)) | |
7 | 6 | rexbidv 2436 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)) |
8 | 5, 7 | elab3 2831 | . 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
9 | 2, 8 | bitri 183 | 1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 {cab 2123 ∃wrex 2415 Vcvv 2681 class class class wbr 3924 ωcom 4499 ≈ cen 6625 Fincfn 6627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-en 6628 df-fin 6630 |
This theorem is referenced by: snfig 6701 fict 6755 fidceq 6756 nnfi 6759 enfi 6760 ssfilem 6762 dif1enen 6767 php5fin 6769 fisbth 6770 fin0 6772 fin0or 6773 diffitest 6774 findcard 6775 findcard2 6776 findcard2s 6777 diffisn 6780 infnfi 6782 fientri3 6796 unsnfi 6800 unsnfidcex 6801 unsnfidcel 6802 fiintim 6810 fidcenumlemim 6833 finnum 7032 hashcl 10520 hashen 10523 fihashdom 10542 hashun 10544 zfz1iso 10577 |
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