Theorem dfom5b 36749

Description: A quantifier-free definition of ω that does not depend on ax-inf 9640. (Note: label was changed from dfom5 9652 to dfom5b 36749 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 4912	. . . . 5 ⊢ (𝑥 ∈ ∩ Limits ↔ ∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦))
3		vex 3454	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	3	ellimits 36747	. . . . . . 7 ⊢ (𝑦 ∈ Limits ↔ Lim 𝑦)
5	4	imbi1i 352	. . . . . 6 ⊢ ((𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ (Lim 𝑦 → 𝑥 ∈ 𝑦))
6	5	albii 1851	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ Limits → 𝑥 ∈ 𝑦) ↔ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
7	2, 6	bitr2i 279	. . . 4 ⊢ (∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ ∩ Limits )
8	7	anbi2i 635	. . 3 ⊢ ((𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits ))
9		elom 7880	. . 3 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)))
10		elin 4093	. . 3 ⊢ (𝑥 ∈ (On ∩ ∩ Limits ) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ ∩ Limits ))
11	8, 9, 10	3bitr4i 306	. 2 ⊢ (𝑥 ∈ ω ↔ 𝑥 ∈ (On ∩ ∩ Limits ))
12	11	eqriv 2757	1 ⊢ ω = (On ∩ ∩ Limits )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144   ∩ cin 4089  ∩ cint 4906  Oncon0 6362  Lim wlim 6363  ωcom 7877   Limits climits 36673

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-symdif 4198  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ord 6365  df-on 6366  df-lim 6367  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-om 7878  df-1st 8001  df-2nd 8002  df-txp 36691  df-bigcup 36695  df-fix 36696  df-limits 36697

This theorem is used by: (None)