Theorem oasuc 6610

Description: Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

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

Proof of Theorem oasuc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsuc 4593	. . . . . 6 |- ( B e. On -> suc B e. On )
2		oav2 6609	. . . . . 6 |- ( ( A e. On /\ suc B e. On ) -> ( A +o suc B ) = ( A u. U_ x e. suc B suc ( A +o x ) ) )
3	1, 2	sylan2 286	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A +o suc B ) = ( A u. U_ x e. suc B suc ( A +o x ) ) )
4		df-suc 4462	. . . . . . . . . 10 |- suc B = ( B u. { B } )
5		iuneq1 3978	. . . . . . . . . 10 |- ( suc B = ( B u. { B } ) -> U_ x e. suc B suc ( A +o x ) = U_ x e. ( B u. { B } ) suc ( A +o x ) )
6	4, 5	ax-mp 5	. . . . . . . . 9 |- U_ x e. suc B suc ( A +o x ) = U_ x e. ( B u. { B } ) suc ( A +o x )
7		iunxun 4045	. . . . . . . . 9 |- U_ x e. ( B u. { B } ) suc ( A +o x ) = ( U_ x e. B suc ( A +o x ) u. U_ x e. { B } suc ( A +o x ) )
8	6, 7	eqtri 2250	. . . . . . . 8 |- U_ x e. suc B suc ( A +o x ) = ( U_ x e. B suc ( A +o x ) u. U_ x e. { B } suc ( A +o x ) )
9		oveq2 6009	. . . . . . . . . . 11 |- ( x = B -> ( A +o x ) = ( A +o B ) )
10		suceq 4493	. . . . . . . . . . 11 |- ( ( A +o x ) = ( A +o B ) -> suc ( A +o x ) = suc ( A +o B ) )
11	9, 10	syl 14	. . . . . . . . . 10 |- ( x = B -> suc ( A +o x ) = suc ( A +o B ) )
12	11	iunxsng 4041	. . . . . . . . 9 |- ( B e. On -> U_ x e. { B } suc ( A +o x ) = suc ( A +o B ) )
13	12	uneq2d 3358	. . . . . . . 8 |- ( B e. On -> ( U_ x e. B suc ( A +o x ) u. U_ x e. { B } suc ( A +o x ) ) = ( U_ x e. B suc ( A +o x ) u. suc ( A +o B ) ) )
14	8, 13	eqtrid 2274	. . . . . . 7 |- ( B e. On -> U_ x e. suc B suc ( A +o x ) = ( U_ x e. B suc ( A +o x ) u. suc ( A +o B ) ) )
15	14	uneq2d 3358	. . . . . 6 |- ( B e. On -> ( A u. U_ x e. suc B suc ( A +o x ) ) = ( A u. ( U_ x e. B suc ( A +o x ) u. suc ( A +o B ) ) ) )
16	15	adantl 277	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A u. U_ x e. suc B suc ( A +o x ) ) = ( A u. ( U_ x e. B suc ( A +o x ) u. suc ( A +o B ) ) ) )
17	3, 16	eqtrd 2262	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A +o suc B ) = ( A u. ( U_ x e. B suc ( A +o x ) u. suc ( A +o B ) ) ) )
18		unass 3361	. . . 4 |- ( ( A u. U_ x e. B suc ( A +o x ) ) u. suc ( A +o B ) ) = ( A u. ( U_ x e. B suc ( A +o x ) u. suc ( A +o B ) ) )
19	17, 18	eqtr4di 2280	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A +o suc B ) = ( ( A u. U_ x e. B suc ( A +o x ) ) u. suc ( A +o B ) ) )
20		oav2 6609	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A +o B ) = ( A u. U_ x e. B suc ( A +o x ) ) )
21	20	uneq1d 3357	. . 3 |- ( ( A e. On /\ B e. On ) -> ( ( A +o B ) u. suc ( A +o B ) ) = ( ( A u. U_ x e. B suc ( A +o x ) ) u. suc ( A +o B ) ) )
22	19, 21	eqtr4d 2265	. 2 |- ( ( A e. On /\ B e. On ) -> ( A +o suc B ) = ( ( A +o B ) u. suc ( A +o B ) ) )
23		sssucid 4506	. . 3 |- ( A +o B ) C_ suc ( A +o B )
24		ssequn1 3374	. . 3 |- ( ( A +o B ) C_ suc ( A +o B ) <-> ( ( A +o B ) u. suc ( A +o B ) ) = suc ( A +o B ) )
25	23, 24	mpbi 145	. 2 |- ( ( A +o B ) u. suc ( A +o B ) ) = suc ( A +o B )
26	22, 25	eqtrdi 2278	1 |- ( ( A e. On /\ B e. On ) -> ( A +o suc B ) = suc ( A +o B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   u. cun 3195   C_ wss 3197   {csn 3666   U_ciun 3965   Oncon0 4454   suc csuc 4456  (class class class)co 6001   +o coa 6559

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-oadd 6566

This theorem is referenced by:  onasuc  6612  nnaordi  6654