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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfom5b | Structured version Visualization version GIF version |
Description: A quantifier-free definition of ω that does not depend on ax-inf 8418. (Note: label was changed from dfom5 8430 to dfom5b 31189 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
dfom5b | ⊢ ω = (On ∩ ∩ Limits ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | elint 4416 | . . . . 5 ⊢ (𝑥 ∈ ∩ Limits ↔ ∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦)) |
3 | vex 3176 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | ellimits 31187 | . . . . . . 7 ⊢ (𝑦 ∈ Limits ↔ Lim 𝑦) |
5 | 4 | imbi1i 338 | . . . . . 6 ⊢ ((𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ (Lim 𝑦 → 𝑥 ∈ 𝑦)) |
6 | 5 | albii 1737 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) |
7 | 2, 6 | bitr2i 264 | . . . 4 ⊢ (∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ ∩ Limits ) |
8 | 7 | anbi2i 726 | . . 3 ⊢ ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits )) |
9 | elom 6960 | . . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))) | |
10 | elin 3758 | . . 3 ⊢ (𝑥 ∈ (On ∩ ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits )) | |
11 | 8, 9, 10 | 3bitr4i 291 | . 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ ∩ Limits )) |
12 | 11 | eqriv 2607 | 1 ⊢ ω = (On ∩ ∩ Limits ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ∩ cint 4410 Oncon0 5640 Lim wlim 5641 ωcom 6957 Limits climits 31112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ord 5643 df-on 5644 df-lim 5645 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-om 6958 df-1st 7059 df-2nd 7060 df-txp 31130 df-bigcup 31134 df-fix 31135 df-limits 31136 |
This theorem is referenced by: (None) |
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