Theorem dfom5 8181

Description: om is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
dfom5	|- om = |^| { x | Lim x }

Proof of Theorem dfom5

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elom3 8179	. . 3 |- ( y e. om <-> A. x ( Lim x -> y e. x ) )
2		vex 3060	. . . 4 |- y e. _V
3	2	elintab 4259	. . 3 |- ( y e. |^| { x | Lim x } <-> A. x ( Lim x -> y e. x ) )
4	1, 3	bitr4i 260	. 2 |- ( y e. om <-> y e. |^| { x | Lim x } )
5	4	eqriv 2459	1 |- om = |^| { x | Lim x }

Syntax hints:   -> wi 4   A.wal 1453   = wceq 1455   e. wcel 1898   {cab 2448   |^|cint 4248   Lim wlim 5443   omcom 6719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653  ax-un 6610  ax-inf2 8172

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-br 4417  df-opab 4476  df-tr 4512  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-om 6720

This theorem is referenced by: (None)