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| Description: A simplification of elom 7891 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.) | 
| Ref | Expression | 
|---|---|
| elom3 | ⊢ (𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elom 7891 | . 2 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
| 2 | limom 7904 | . . . . 5 ⊢ Lim ω | |
| 3 | omex 9684 | . . . . . 6 ⊢ ω ∈ V | |
| 4 | limeq 6395 | . . . . . . 7 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω)) | |
| 5 | eleq2 2829 | . . . . . . 7 ⊢ (𝑥 = ω → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ω)) | |
| 6 | 4, 5 | imbi12d 344 | . . . . . 6 ⊢ (𝑥 = ω → ((Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (Lim ω → 𝐴 ∈ ω))) | 
| 7 | 3, 6 | spcv 3604 | . . . . 5 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) → (Lim ω → 𝐴 ∈ ω)) | 
| 8 | 2, 7 | mpi 20 | . . . 4 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) → 𝐴 ∈ ω) | 
| 9 | nnon 7894 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
| 10 | 8, 9 | syl 17 | . . 3 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) → 𝐴 ∈ On) | 
| 11 | 10 | pm4.71ri 560 | . 2 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | 
| 12 | 1, 11 | bitr4i 278 | 1 ⊢ (𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 Oncon0 6383 Lim wlim 6384 ωcom 7888 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-inf2 9682 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-om 7889 | 
| This theorem is referenced by: dfom4 9690 dfom5 9691 | 
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