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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 8501 | . . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} | |
2 | 1 | eleq2i 2901 | . 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}) |
3 | relen 8502 | . . . . 5 ⊢ Rel ≈ | |
4 | 3 | brrelex1i 5601 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
5 | 4 | rexlimivw 3279 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
6 | breq1 5060 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥)) | |
7 | 6 | rexbidv 3294 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)) |
8 | 5, 7 | elab3 3671 | . 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
9 | 2, 8 | bitri 276 | 1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 {cab 2796 ∃wrex 3136 Vcvv 3492 class class class wbr 5057 ωcom 7569 ≈ cen 8494 Fincfn 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-en 8498 df-fin 8501 |
This theorem is referenced by: snfi 8582 php3 8691 onfin 8697 ominf 8718 isinf 8719 enfi 8722 ssnnfi 8725 ssfi 8726 dif1en 8739 findcard 8745 findcard2 8746 findcard3 8749 nnsdomg 8765 isfiniteg 8766 unfi 8773 fiint 8783 pwfi 8807 finnum 9365 ficardom 9378 dif1card 9424 infpwfien 9476 ficard 9975 hashkf 13680 finminlem 33563 domalom 34567 |
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