Theorem dfom5b 35895

Description: A quantifier-free definition of ω that does not depend on ax-inf 9597. (Note: label was changed from dfom5 9609 to dfom5b 35895 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.

Step	Hyp	Ref	Expression
1		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	elint 4918	. . . . 5 ⊢ (𝑥 ∈ ∩ Limits ↔ ∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦))
3		vex 3454	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	3	ellimits 35893	. . . . . . 7 ⊢ (𝑦 ∈ Limits ↔ Lim 𝑦)
5	4	imbi1i 349	. . . . . 6 ⊢ ((𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ (Lim 𝑦 → 𝑥 ∈ 𝑦))
6	5	albii 1819	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
7	2, 6	bitr2i 276	. . . 4 ⊢ (∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ ∩ Limits )
8	7	anbi2i 623	. . 3 ⊢ ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits ))
9		elom 7847	. . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)))
10		elin 3932	. . 3 ⊢ (𝑥 ∈ (On ∩ ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits ))
11	8, 9, 10	3bitr4i 303	. 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ ∩ Limits ))
12	11	eqriv 2727	1 ⊢ ω = (On ∩ ∩ Limits )

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ∩ cin 3915  ∩ cint 4912  Oncon0 6334  Lim wlim 6335  ωcom 7844   Limits climits 35819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-symdif 4218  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ord 6337  df-on 6338  df-lim 6339  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fo 6519  df-fv 6521  df-om 7845  df-1st 7970  df-2nd 7971  df-txp 35837  df-bigcup 35841  df-fix 35842  df-limits 35843

This theorem is referenced by: (None)