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Mirrors > Home > MPE Home > Th. List > elom3 | Structured version Visualization version GIF version |
Description: A simplification of elom 7348 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 7348 | . 2 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
2 | limom 7360 | . . . . 5 ⊢ Lim ω | |
3 | omex 8839 | . . . . . 6 ⊢ ω ∈ V | |
4 | limeq 5990 | . . . . . . 7 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω)) | |
5 | eleq2 2848 | . . . . . . 7 ⊢ (𝑥 = ω → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ω)) | |
6 | 4, 5 | imbi12d 336 | . . . . . 6 ⊢ (𝑥 = ω → ((Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (Lim ω → 𝐴 ∈ ω))) |
7 | 3, 6 | spcv 3501 | . . . . 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 7351 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) → 𝐴 ∈ On) |
11 | 10 | pm4.71ri 556 | . 2 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
12 | 1, 11 | bitr4i 270 | 1 ⊢ (𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∀wal 1599 = wceq 1601 ∈ wcel 2107 Oncon0 5978 Lim wlim 5979 ωcom 7345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 ax-inf2 8837 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-tr 4990 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-om 7346 |
This theorem is referenced by: dfom4 8845 dfom5 8846 |
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