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Mirrors > Home > MPE Home > Th. List > isfin4 | Structured version Visualization version GIF version |
Description: Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
isfin4 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psseq2 3695 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ⊊ 𝑥 ↔ 𝑦 ⊊ 𝐴)) | |
2 | breq2 4657 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴)) | |
3 | 1, 2 | anbi12d 747 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ (𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴))) |
4 | 3 | exbidv 1850 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴))) |
5 | 4 | notbid 308 | . 2 ⊢ (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴))) |
6 | df-fin4 9109 | . 2 ⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)} | |
7 | 5, 6 | elab2g 3353 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ⊊ wpss 3575 class class class wbr 4653 ≈ cen 7952 FinIVcfin4 9102 |
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 |
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-ne 2795 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-fin4 9109 |
This theorem is referenced by: fin4i 9120 fin4en1 9131 ssfin4 9132 infpssALT 9135 isfin4-2 9136 |
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