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Mirrors > Home > MPE Home > Th. List > elom | Structured version Visualization version GIF version |
Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9686. (Contributed by NM, 15-May-1994.) |
Ref | Expression |
---|---|
elom | ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2827 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
2 | 1 | imbi2d 340 | . . 3 ⊢ (𝑦 = 𝐴 → ((Lim 𝑥 → 𝑦 ∈ 𝑥) ↔ (Lim 𝑥 → 𝐴 ∈ 𝑥))) |
3 | 2 | albidv 1918 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
4 | df-om 7888 | . 2 ⊢ ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥)} | |
5 | 3, 4 | elrab2 3698 | 1 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 Oncon0 6386 Lim wlim 6387 ωcom 7887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-om 7888 |
This theorem is referenced by: limomss 7892 trom 7896 nnlim 7901 limom 7903 peano1 7911 1onn 8677 2onn 8679 elom3 9686 dfom5b 35894 |
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