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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 7959 | . . 3 ⊢ Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} | |
2 | 1 | eleq2i 2693 | . 2 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥}) |
3 | relen 7960 | . . . . 5 ⊢ Rel ≈ | |
4 | 3 | brrelexi 5158 | . . . 4 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
5 | 4 | rexlimivw 3029 | . . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
6 | breq1 4656 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝐴 ≈ 𝑥)) | |
7 | 6 | rexbidv 3052 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)) |
8 | 5, 7 | elab3 3358 | . 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦 ≈ 𝑥} ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
9 | 2, 8 | bitri 264 | 1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 Vcvv 3200 class class class wbr 4653 ωcom 7065 ≈ cen 7952 Fincfn 7955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-en 7956 df-fin 7959 |
This theorem is referenced by: snfi 8038 php3 8146 onfin 8151 ominf 8172 isinf 8173 enfi 8176 ssnnfi 8179 ssfi 8180 dif1en 8193 findcard 8199 findcard2 8200 findcard3 8203 nnsdomg 8219 isfiniteg 8220 unfi 8227 fiint 8237 pwfi 8261 finnum 8774 ficardom 8787 dif1card 8833 infpwfien 8885 ficard 9387 hashkf 13119 finminlem 32312 |
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