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Mirrors > Home > MPE Home > Th. List > isfin7 | Structured version Visualization version GIF version |
Description: Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
isfin7 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5030 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝐴 ≈ 𝑦)) | |
2 | 1 | rexbidv 3206 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦)) |
3 | 2 | notbid 321 | . 2 ⊢ (𝑥 = 𝐴 → (¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦 ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦)) |
4 | df-fin7 9784 | . 2 ⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦} | |
5 | 3, 4 | elab2g 3572 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2113 ∃wrex 3054 ∖ cdif 3838 class class class wbr 5027 Oncon0 6166 ωcom 7593 ≈ cen 8545 FinVIIcfin7 9777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-rex 3059 df-v 3399 df-un 3846 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-fin7 9784 |
This theorem is referenced by: fin17 9887 fin67 9888 isfin7-2 9889 |
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