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| Mirrors > Home > MPE Home > Th. List > isfi | Structured version Visualization version 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 8989 | . . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}) |
| 3 | relen 8990 | . . . . 5 ⊢ Rel ≈ | |
| 4 | 3 | brrelex1i 5741 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
| 5 | 4 | rexlimivw 3151 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
| 6 | breq1 5146 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥)) | |
| 7 | 6 | rexbidv 3179 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)) |
| 8 | 5, 7 | elab3 3686 | . 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
| 9 | 2, 8 | bitri 275 | 1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 class class class wbr 5143 ωcom 7887 ≈ cen 8982 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-en 8986 df-fin 8989 |
| This theorem is referenced by: 0fi 9082 snfi 9083 snfiOLD 9084 findcard 9203 findcard2 9204 nnfi 9207 ssnnfi 9209 unfi 9211 ssfiALT 9214 enfii 9226 enfiALT 9228 php3 9249 php3OLD 9261 onfin 9267 ominf 9294 ominfOLD 9295 isinf 9296 isinfOLD 9297 dif1ennnALT 9311 findcard3 9318 findcard3OLD 9319 nnsdomg 9335 nnsdomgOLD 9336 isfiniteg 9337 prfi 9363 fiint 9366 fiintOLD 9367 finnum 9988 ficardom 10001 dif1card 10050 infpwfien 10102 ficard 10605 hashkf 14371 finminlem 36319 domalom 37405 |
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