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Mirrors > Home > MPE Home > Th. List > isfin6 | Structured version Visualization version GIF version |
Description: Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
isfin6 | ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fin6 9712 | . . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))}) |
3 | relsdom 8516 | . . . . 5 ⊢ Rel ≺ | |
4 | 3 | brrelex1i 5608 | . . . 4 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ V) |
5 | 3 | brrelex1i 5608 | . . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V) |
6 | 4, 5 | jaoi 853 | . . 3 ⊢ ((𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V) |
7 | breq1 5069 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2o ↔ 𝐴 ≺ 2o)) | |
8 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
9 | 8 | sqxpeqd 5587 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴)) |
10 | 8, 9 | breq12d 5079 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴))) |
11 | 7, 10 | orbi12d 915 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)))) |
12 | 6, 11 | elab3 3674 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
13 | 2, 12 | bitri 277 | 1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cab 2799 Vcvv 3494 class class class wbr 5066 × cxp 5553 2oc2o 8096 ≺ csdm 8508 FinVIcfin6 9705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-dom 8511 df-sdom 8512 df-fin6 9712 |
This theorem is referenced by: fin56 9815 fin67 9817 |
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