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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 6224 | . . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} | |
2 | 1 | eleq2i 2104 | . 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}) |
3 | relen 6225 | . . . . 5 ⊢ Rel ≈ | |
4 | 3 | brrelexi 4384 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
5 | 4 | rexlimivw 2429 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
6 | breq1 3767 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥)) | |
7 | 6 | rexbidv 2327 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)) |
8 | 5, 7 | elab3 2694 | . 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
9 | 2, 8 | bitri 173 | 1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 ∈ wcel 1393 {cab 2026 ∃wrex 2307 Vcvv 2557 class class class wbr 3764 ωcom 4313 ≈ cen 6219 Fincfn 6221 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-en 6222 df-fin 6224 |
This theorem is referenced by: snfig 6291 fidceq 6330 nnfi 6333 enfi 6334 ssfiexmid 6336 php5fin 6339 fisbth 6340 fin0 6342 fin0or 6343 diffitest 6344 findcard 6345 findcard2 6346 findcard2s 6347 diffisn 6350 fientri3 6353 finnum 6363 |
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