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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 9089. (Note: label was changed from dfom5 9101 to dfom5b 33270 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 3495 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | elint 4873 | . . . . 5 ⊢ (𝑥 ∈ ∩ Limits ↔ ∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦)) |
3 | vex 3495 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | ellimits 33268 | . . . . . . 7 ⊢ (𝑦 ∈ Limits ↔ Lim 𝑦) |
5 | 4 | imbi1i 351 | . . . . . 6 ⊢ ((𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ (Lim 𝑦 → 𝑥 ∈ 𝑦)) |
6 | 5 | albii 1811 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) |
7 | 2, 6 | bitr2i 277 | . . . 4 ⊢ (∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ ∩ Limits ) |
8 | 7 | anbi2i 622 | . . 3 ⊢ ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits )) |
9 | elom 7572 | . . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))) | |
10 | elin 4166 | . . 3 ⊢ (𝑥 ∈ (On ∩ ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits )) | |
11 | 8, 9, 10 | 3bitr4i 304 | . 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ ∩ Limits )) |
12 | 11 | eqriv 2815 | 1 ⊢ ω = (On ∩ ∩ Limits ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ∩ cint 4867 Oncon0 6184 Lim wlim 6185 ωcom 7569 Limits climits 33194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-symdif 4216 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ord 6187 df-on 6188 df-lim 6189 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-om 7570 df-1st 7678 df-2nd 7679 df-txp 33212 df-bigcup 33216 df-fix 33217 df-limits 33218 |
This theorem is referenced by: (None) |
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