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Mirrors > Home > MPE Home > Th. List > elom3 | Structured version Visualization version GIF version |
Description: A simplification of elom 7873 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 7873 | . 2 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))) | |
2 | limom 7886 | . . . . 5 ⊢ Lim ω | |
3 | omex 9667 | . . . . . 6 ⊢ ω ∈ V | |
4 | limeq 6381 | . . . . . . 7 ⊢ (𝑥 = ω → (Lim 𝑥 ↔ Lim ω)) | |
5 | eleq2 2818 | . . . . . . 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 3592 | . . . . 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 7876 | . . . 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 205 ∧ wa 395 ∀wal 1532 = wceq 1534 ∈ wcel 2099 Oncon0 6369 Lim wlim 6370 ωcom 7870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 ax-inf2 9665 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-om 7871 |
This theorem is referenced by: dfom4 9673 dfom5 9674 |
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