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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 9642. (Contributed by NM, 15-May-1994.) |
Ref | Expression |
---|---|
elom | ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2821 | . . . 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 1923 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
4 | df-om 7855 | . 2 ⊢ ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥)} | |
5 | 3, 4 | elrab2 3686 | 1 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 Oncon0 6364 Lim wlim 6365 ωcom 7854 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-om 7855 |
This theorem is referenced by: limomss 7859 trom 7863 nnlim 7868 limom 7870 peano1 7878 1onn 8638 2onn 8640 elom3 9642 dfom5b 34879 |
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