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Theorem dfom5b 32525
Description: A quantifier-free definition of ω that does not depend on ax-inf 8784. (Note: label was changed from dfom5 8796 to dfom5b 32525 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)
Assertion
Ref Expression
dfom5b ω = (On ∩ Limits )

Proof of Theorem dfom5b
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3387 . . . . . 6 𝑥 ∈ V
21elint 4672 . . . . 5 (𝑥 Limits ↔ ∀𝑦(𝑦 Limits 𝑥𝑦))
3 vex 3387 . . . . . . . 8 𝑦 ∈ V
43ellimits 32523 . . . . . . 7 (𝑦 Limits ↔ Lim 𝑦)
54imbi1i 341 . . . . . 6 ((𝑦 Limits 𝑥𝑦) ↔ (Lim 𝑦𝑥𝑦))
65albii 1915 . . . . 5 (∀𝑦(𝑦 Limits 𝑥𝑦) ↔ ∀𝑦(Lim 𝑦𝑥𝑦))
72, 6bitr2i 268 . . . 4 (∀𝑦(Lim 𝑦𝑥𝑦) ↔ 𝑥 Limits )
87anbi2i 617 . . 3 ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 Limits ))
9 elom 7301 . . 3 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
10 elin 3993 . . 3 (𝑥 ∈ (On ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 Limits ))
118, 9, 103bitr4i 295 . 2 (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ Limits ))
1211eqriv 2795 1 ω = (On ∩ Limits )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651   = wceq 1653  wcel 2157  cin 3767   cint 4666  Oncon0 5940  Lim wlim 5941  ωcom 7298   Limits climits 32449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-symdif 4040  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-int 4667  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ord 5943  df-on 5944  df-lim 5945  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-fo 6106  df-fv 6108  df-om 7299  df-1st 7400  df-2nd 7401  df-txp 32467  df-bigcup 32471  df-fix 32472  df-limits 32473
This theorem is referenced by: (None)
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