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Theorem dfom5b 34815
Description: A quantifier-free definition of ω that does not depend on ax-inf 9620. (Note: label was changed from dfom5 9632 to dfom5b 34815 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 3479 . . . . . 6 𝑥 ∈ V
21elint 4952 . . . . 5 (𝑥 Limits ↔ ∀𝑦(𝑦 Limits 𝑥𝑦))
3 vex 3479 . . . . . . . 8 𝑦 ∈ V
43ellimits 34813 . . . . . . 7 (𝑦 Limits ↔ Lim 𝑦)
54imbi1i 350 . . . . . 6 ((𝑦 Limits 𝑥𝑦) ↔ (Lim 𝑦𝑥𝑦))
65albii 1822 . . . . 5 (∀𝑦(𝑦 Limits 𝑥𝑦) ↔ ∀𝑦(Lim 𝑦𝑥𝑦))
72, 6bitr2i 276 . . . 4 (∀𝑦(Lim 𝑦𝑥𝑦) ↔ 𝑥 Limits )
87anbi2i 624 . . 3 ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 Limits ))
9 elom 7845 . . 3 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
10 elin 3962 . . 3 (𝑥 ∈ (On ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 Limits ))
118, 9, 103bitr4i 303 . 2 (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ Limits ))
1211eqriv 2730 1 ω = (On ∩ Limits )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  cin 3945   cint 4946  Oncon0 6356  Lim wlim 6357  ωcom 7842   Limits climits 34739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-symdif 4240  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ord 6359  df-on 6360  df-lim 6361  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fo 6541  df-fv 6543  df-om 7843  df-1st 7962  df-2nd 7963  df-txp 34757  df-bigcup 34761  df-fix 34762  df-limits 34763
This theorem is referenced by: (None)
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