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Theorem dfom5b 32356
Description: A quantifier-free definition of ω that does not depend on ax-inf 8699. (Note: label was changed from dfom5 8711 to dfom5b 32356 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 3354 . . . . . 6 𝑥 ∈ V
21elint 4617 . . . . 5 (𝑥 Limits ↔ ∀𝑦(𝑦 Limits 𝑥𝑦))
3 vex 3354 . . . . . . . 8 𝑦 ∈ V
43ellimits 32354 . . . . . . 7 (𝑦 Limits ↔ Lim 𝑦)
54imbi1i 338 . . . . . 6 ((𝑦 Limits 𝑥𝑦) ↔ (Lim 𝑦𝑥𝑦))
65albii 1895 . . . . 5 (∀𝑦(𝑦 Limits 𝑥𝑦) ↔ ∀𝑦(Lim 𝑦𝑥𝑦))
72, 6bitr2i 265 . . . 4 (∀𝑦(Lim 𝑦𝑥𝑦) ↔ 𝑥 Limits )
87anbi2i 609 . . 3 ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 Limits ))
9 elom 7215 . . 3 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦𝑥𝑦)))
10 elin 3947 . . 3 (𝑥 ∈ (On ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 Limits ))
118, 9, 103bitr4i 292 . 2 (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ Limits ))
1211eqriv 2768 1 ω = (On ∩ Limits )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629   = wceq 1631  wcel 2145  cin 3722   cint 4611  Oncon0 5866  Lim wlim 5867  ωcom 7212   Limits climits 32280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3993  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ord 5869  df-on 5870  df-lim 5871  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-om 7213  df-1st 7315  df-2nd 7316  df-txp 32298  df-bigcup 32302  df-fix 32303  df-limits 32304
This theorem is referenced by: (None)
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