Theorem dfom5b 32019

Description: A quantifier-free definition of om that does not depend on ax-inf 8535. (Note: label was changed from dfom5 8547 to dfom5b 32019 to prevent naming conflict. NM, 12-Feb-2013.) (Contributed by Scott Fenton, 11-Apr-2012.)

Assertion

Ref	Expression
dfom5b	|- om = ( On i^i |^| Limits )

Proof of Theorem dfom5b

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- x e. _V
2	1	elint 4481	. . . . 5 |- ( x e. |^| Limits <-> A. y ( y e. Limits -> x e. y ) )
3		vex 3203	. . . . . . . 8 |- y e. _V
4	3	ellimits 32017	. . . . . . 7 |- ( y e. Limits <-> Lim y )
5	4	imbi1i 339	. . . . . 6 |- ( ( y e. Limits -> x e. y ) <-> ( Lim y -> x e. y ) )
6	5	albii 1747	. . . . 5 |- ( A. y ( y e. Limits -> x e. y ) <-> A. y ( Lim y -> x e. y ) )
7	2, 6	bitr2i 265	. . . 4 |- ( A. y ( Lim y -> x e. y ) <-> x e. |^| Limits )
8	7	anbi2i 730	. . 3 |- ( ( x e. On /\ A. y ( Lim y -> x e. y ) ) <-> ( x e. On /\ x e. |^| Limits ) )
9		elom 7068	. . 3 |- ( x e. om <-> ( x e. On /\ A. y ( Lim y -> x e. y ) ) )
10		elin 3796	. . 3 |- ( x e. ( On i^i |^| Limits ) <-> ( x e. On /\ x e. |^| Limits ) )
11	8, 9, 10	3bitr4i 292	. 2 |- ( x e. om <-> x e. ( On i^i |^| Limits ) )
12	11	eqriv 2619	1 |- om = ( On i^i |^| Limits )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   i^i cin 3573   |^|cint 4475   Oncon0 5723   Lim wlim 5724   omcom 7065   Limitsclimits 31943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ord 5726  df-on 5727  df-lim 5728  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-om 7066  df-1st 7168  df-2nd 7169  df-txp 31961  df-bigcup 31965  df-fix 31966  df-limits 31967

This theorem is referenced by: (None)