Theorem oacl 6671

Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)

Assertion

Ref	Expression
oacl	|- ( ( A e. On /\ B e. On ) -> ( A +o B ) e. On )

Proof of Theorem oacl

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oav 6665	. 2 |- ( ( A e. On /\ B e. On ) -> ( A +o B ) = ( rec ( ( z e. _V |-> suc z ) , A ) ` B ) )
2		id 19	. . 3 |- ( A e. On -> A e. On )
3		vex 2806	. . . . . . . 8 |- w e. _V
4		suceq 4505	. . . . . . . . 9 |- ( z = w -> suc z = suc w )
5		eqid 2231	. . . . . . . . 9 |- ( z e. _V |-> suc z ) = ( z e. _V |-> suc z )
6	3	sucex 4603	. . . . . . . . 9 |- suc w e. _V
7	4, 5, 6	fvmpt 5732	. . . . . . . 8 |- ( w e. _V -> ( ( z e. _V |-> suc z ) ` w ) = suc w )
8	3, 7	ax-mp 5	. . . . . . 7 |- ( ( z e. _V |-> suc z ) ` w ) = suc w
9	8	eleq1i 2297	. . . . . 6 |- ( ( ( z e. _V |-> suc z ) ` w ) e. On <-> suc w e. On )
10	9	ralbii 2539	. . . . 5 |- ( A. w e. On ( ( z e. _V |-> suc z ) ` w ) e. On <-> A. w e. On suc w e. On )
11		onsuc 4605	. . . . 5 |- ( w e. On -> suc w e. On )
12	10, 11	mprgbir 2591	. . . 4 |- A. w e. On ( ( z e. _V |-> suc z ) ` w ) e. On
13	12	a1i 9	. . 3 |- ( A e. On -> A. w e. On ( ( z e. _V |-> suc z ) ` w ) e. On )
14	2, 13	rdgon 6595	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( z e. _V |-> suc z ) , A ) ` B ) e. On )
15	1, 14	eqeltrd 2308	1 |- ( ( A e. On /\ B e. On ) -> ( A +o B ) e. On )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   |-> cmpt 4155   Oncon0 4466   suc csuc 4468   ` cfv 5333  (class class class)co 6028   reccrdg 6578   +o coa 6622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-recs 6514  df-irdg 6579  df-oadd 6629

This theorem is referenced by:  omcl  6672  omv2  6676