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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 9231. (Note: label was changed from dfom5 9243 to dfom5b 33900 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 3402 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | elint 4851 | . . . . 5 ⊢ (𝑥 ∈ ∩ Limits ↔ ∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦)) |
3 | vex 3402 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | ellimits 33898 | . . . . . . 7 ⊢ (𝑦 ∈ Limits ↔ Lim 𝑦) |
5 | 4 | imbi1i 353 | . . . . . 6 ⊢ ((𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ (Lim 𝑦 → 𝑥 ∈ 𝑦)) |
6 | 5 | albii 1827 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) |
7 | 2, 6 | bitr2i 279 | . . . 4 ⊢ (∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ ∩ Limits ) |
8 | 7 | anbi2i 626 | . . 3 ⊢ ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits )) |
9 | elom 7625 | . . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))) | |
10 | elin 3869 | . . 3 ⊢ (𝑥 ∈ (On ∩ ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits )) | |
11 | 8, 9, 10 | 3bitr4i 306 | . 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ ∩ Limits )) |
12 | 11 | eqriv 2733 | 1 ⊢ ω = (On ∩ ∩ Limits ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 ∩ cint 4845 Oncon0 6191 Lim wlim 6192 ωcom 7622 Limits climits 33824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-symdif 4143 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ord 6194 df-on 6195 df-lim 6196 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-om 7623 df-1st 7739 df-2nd 7740 df-txp 33842 df-bigcup 33846 df-fix 33847 df-limits 33848 |
This theorem is referenced by: (None) |
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