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Mirrors > Home > MPE Home > Th. List > isfin5 | Structured version Visualization version GIF version |
Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
isfin5 | ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fin5 8994 | . . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))} | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))}) |
3 | id 22 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
4 | 0ex 4718 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | syl6eqel 2696 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V) |
6 | relsdom 7848 | . . . . 5 ⊢ Rel ≺ | |
7 | 6 | brrelexi 5082 | . . . 4 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V) |
8 | 5, 7 | jaoi 393 | . . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)) → 𝐴 ∈ V) |
9 | eqeq1 2614 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
10 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
11 | 10, 10 | oveq12d 6567 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +𝑐 𝑥) = (𝐴 +𝑐 𝐴)) |
12 | 10, 11 | breq12d 4596 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 +𝑐 𝑥) ↔ 𝐴 ≺ (𝐴 +𝑐 𝐴))) |
13 | 9, 12 | orbi12d 742 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))) |
14 | 8, 13 | elab3 3327 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴))) |
15 | 2, 14 | bitri 263 | 1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 ∅c0 3874 class class class wbr 4583 (class class class)co 6549 ≺ csdm 7840 +𝑐 ccda 8872 FinVcfin5 8987 |
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-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-iota 5768 df-fv 5812 df-ov 6552 df-dom 7843 df-sdom 7844 df-fin5 8994 |
This theorem is referenced by: isfin5-2 9096 fin56 9098 |
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