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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 7845 | . . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}) |
3 | relen 7846 | . . . . 5 ⊢ Rel ≈ | |
4 | 3 | brrelexi 5082 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
5 | 4 | rexlimivw 3011 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
6 | breq1 4586 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥)) | |
7 | 6 | rexbidv 3034 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)) |
8 | 5, 7 | elab3 3327 | . 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
9 | 2, 8 | bitri 263 | 1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 ωcom 6957 ≈ cen 7838 Fincfn 7841 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-en 7842 df-fin 7845 |
This theorem is referenced by: snfi 7923 php3 8031 onfin 8036 ominf 8057 isinf 8058 enfi 8061 ssnnfi 8064 ssfi 8065 dif1en 8078 findcard 8084 findcard2 8085 findcard3 8088 nnsdomg 8104 isfiniteg 8105 unfi 8112 fiint 8122 pwfi 8144 finnum 8657 ficardom 8670 dif1card 8716 infpwfien 8768 ficard 9266 hashkf 12981 finminlem 31482 |
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