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Mirrors > Home > MPE Home > Th. List > elom3 | Structured version Visualization version GIF version |
Description: A simplification of elom 7577 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 7577 | . 2 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
2 | limom 7589 | . . . . 5 ⊢ Lim ω | |
3 | omex 9100 | . . . . . 6 ⊢ ω ∈ V | |
4 | limeq 6197 | . . . . . . 7 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω)) | |
5 | eleq2 2901 | . . . . . . 7 ⊢ (𝑥 = ω → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ω)) | |
6 | 4, 5 | imbi12d 347 | . . . . . 6 ⊢ (𝑥 = ω → ((Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (Lim ω → 𝐴 ∈ ω))) |
7 | 3, 6 | spcv 3605 | . . . . 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 7580 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) → 𝐴 ∈ On) |
11 | 10 | pm4.71ri 563 | . 2 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
12 | 1, 11 | bitr4i 280 | 1 ⊢ (𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 Oncon0 6185 Lim wlim 6186 ωcom 7574 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-om 7575 |
This theorem is referenced by: dfom4 9106 dfom5 9107 |
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