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Theorem dfom5b 32325
Description: A quantifier-free definition of ω that does not depend on ax-inf 8708. (Note: label was changed from dfom5 8720 to dfom5b 32325 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 3343 . . . . . 6 𝑥 ∈ V
21elint 4633 . . . . 5 (𝑥 Limits ↔ ∀𝑦(𝑦 Limits 𝑥𝑦))
3 vex 3343 . . . . . . . 8 𝑦 ∈ V
43ellimits 32323 . . . . . . 7 (𝑦 Limits ↔ Lim 𝑦)
54imbi1i 338 . . . . . 6 ((𝑦 Limits 𝑥𝑦) ↔ (Lim 𝑦𝑥𝑦))
65albii 1896 . . . . 5 (∀𝑦(𝑦 Limits 𝑥𝑦) ↔ ∀𝑦(Lim 𝑦𝑥𝑦))
72, 6bitr2i 265 . . . 4 (∀𝑦(Lim 𝑦𝑥𝑦) ↔ 𝑥 Limits )
87anbi2i 732 . . 3 ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 Limits ))
9 elom 7233 . . 3 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
10 elin 3939 . . 3 (𝑥 ∈ (On ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 Limits ))
118, 9, 103bitr4i 292 . 2 (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ Limits ))
1211eqriv 2757 1 ω = (On ∩ Limits )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630   = wceq 1632  wcel 2139  cin 3714   cint 4627  Oncon0 5884  Lim wlim 5885  ωcom 7230   Limits climits 32249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-symdif 3987  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ord 5887  df-on 5888  df-lim 5889  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-om 7231  df-1st 7333  df-2nd 7334  df-txp 32267  df-bigcup 32271  df-fix 32272  df-limits 32273
This theorem is referenced by: (None)
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