Theorem dfom5b 34915

Description: A quantifier-free definition of ω that does not depend on ax-inf 9633. (Note: label was changed from dfom5 9645 to dfom5b 34915 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 3479	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	elint 4957	. . . . 5 ⊢ (𝑥 ∈ ∩ Limits ↔ ∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦))
3		vex 3479	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	3	ellimits 34913	. . . . . . 7 ⊢ (𝑦 ∈ Limits ↔ Lim 𝑦)
5	4	imbi1i 350	. . . . . 6 ⊢ ((𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ (Lim 𝑦 → 𝑥 ∈ 𝑦))
6	5	albii 1822	. . . . 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 624	. . 3 ⊢ ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits ))
9		elom 7858	. . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)))
10		elin 3965	. . 3 ⊢ (𝑥 ∈ (On ∩ ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits ))
11	8, 9, 10	3bitr4i 303	. 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ ∩ Limits ))
12	11	eqriv 2730	1 ⊢ ω = (On ∩ ∩ Limits )

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1540   = wceq 1542   ∈ wcel 2107   ∩ cin 3948  ∩ cint 4951  Oncon0 6365  Lim wlim 6366  ωcom 7855   Limits climits 34839

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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4243  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ord 6368  df-on 6369  df-lim 6370  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-om 7856  df-1st 7975  df-2nd 7976  df-txp 34857  df-bigcup 34861  df-fix 34862  df-limits 34863

This theorem is referenced by: (None)