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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 5033 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝐴 ≈ 𝑦)) | |
2 | 1 | rexbidv 3256 | . . 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 9702 | . 2 ⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦} | |
5 | 3, 4 | elab2g 3616 | 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 1538 ∈ wcel 2111 ∃wrex 3107 ∖ cdif 3878 class class class wbr 5030 Oncon0 6159 ωcom 7560 ≈ cen 8489 FinVIIcfin7 9695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rex 3112 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-fin7 9702 |
This theorem is referenced by: fin17 9805 fin67 9806 isfin7-2 9807 |
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