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Mirrors > Home > MPE Home > Th. List > elom3 | Structured version Visualization version GIF version |
Description: A simplification of elom 7854 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 7854 | . 2 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
2 | limom 7867 | . . . . 5 ⊢ Lim ω | |
3 | omex 9634 | . . . . . 6 ⊢ ω ∈ V | |
4 | limeq 6373 | . . . . . . 7 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω)) | |
5 | eleq2 2822 | . . . . . . 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 3595 | . . . . 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 7857 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) → 𝐴 ∈ On) |
11 | 10 | pm4.71ri 561 | . 2 ⊢ (∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥) ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) |
12 | 1, 11 | bitr4i 277 | 1 ⊢ (𝐴 ∈ ω ↔ ∀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 6361 Lim wlim 6362 ωcom 7851 |
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 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 ax-inf2 9632 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-om 7852 |
This theorem is referenced by: dfom4 9640 dfom5 9641 |
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