Theorem tfinsuc 4499

Description: The finite T operator over a successor. (Contributed by SF, 30-Jan-2015.)

Assertion

Ref	Expression
tfinsuc	|- ((A e. Nn /\ (A +o 1c) =/= (/)) -> Tfin (A +o 1c) = ( Tfin A +o 1c))

Proof of Theorem tfinsuc

Dummy variables a b x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3560	. . 3 |- ((A +o 1c) =/= (/) <-> E.a a e. (A +o 1c))
2		peano2 4404	. . . . . . . 8 |- (A e. Nn -> (A +o 1c) e. Nn )
3		tfincl 4493	. . . . . . . 8 |- ((A +o 1c) e. Nn -> Tfin (A +o 1c) e. Nn )
4	2, 3	syl 15	. . . . . . 7 |- (A e. Nn -> Tfin (A +o 1c) e. Nn )
5	4	adantr 451	. . . . . 6 |- ((A e. Nn /\ a e. (A +o 1c)) -> Tfin (A +o 1c) e. Nn )
6		tfincl 4493	. . . . . . . 8 |- (A e. Nn -> Tfin A e. Nn )
7		peano2 4404	. . . . . . . 8 |- ( Tfin A e. Nn -> ( Tfin A +o 1c) e. Nn )
8	6, 7	syl 15	. . . . . . 7 |- (A e. Nn -> ( Tfin A +o 1c) e. Nn )
9	8	adantr 451	. . . . . 6 |- ((A e. Nn /\ a e. (A +o 1c)) -> ( Tfin A +o 1c) e. Nn )
10		tfinpw1 4495	. . . . . . 7 |- (((A +o 1c) e. Nn /\ a e. (A +o 1c)) -> ~P1 a e. Tfin (A +o 1c))
11	2, 10	sylan 457	. . . . . 6 |- ((A e. Nn /\ a e. (A +o 1c)) -> ~P1 a e. Tfin (A +o 1c))
12		elsuc 4414	. . . . . . . 8 |- (a e. (A +o 1c) <-> E.b e. A E.x e. ∼ ba = (b u. {x}))
13		tfinpw1 4495	. . . . . . . . . . . 12 |- ((A e. Nn /\ b e. A) -> ~P1 b e. Tfin A)
14	13	adantrr 697	. . . . . . . . . . 11 |- ((A e. Nn /\ (b e. A /\ x e. ∼ b)) -> ~P1 b e. Tfin A)
15		vex 2863	. . . . . . . . . . . . . . 15 |- x e. _V
16	15	elcompl 3226	. . . . . . . . . . . . . 14 |- (x e. ∼ b <-> -. x e. b)
17		snelpw1 4147	. . . . . . . . . . . . . 14 |- ({x} e. ~P1 b <-> x e. b)
18	16, 17	xchbinxr 302	. . . . . . . . . . . . 13 |- (x e. ∼ b <-> -. {x} e. ~P1 b)
19	18	biimpi 186	. . . . . . . . . . . 12 |- (x e. ∼ b -> -. {x} e. ~P1 b)
20	19	ad2antll 709	. . . . . . . . . . 11 |- ((A e. Nn /\ (b e. A /\ x e. ∼ b)) -> -. {x} e. ~P1 b)
21		snex 4112	. . . . . . . . . . . 12 |- {x} e. _V
22	21	elsuci 4415	. . . . . . . . . . 11 |- ((~P1 b e. Tfin A /\ -. {x} e. ~P1 b) -> (~P1 b u. {{x}}) e. ( Tfin A +o 1c))
23	14, 20, 22	syl2anc 642	. . . . . . . . . 10 |- ((A e. Nn /\ (b e. A /\ x e. ∼ b)) -> (~P1 b u. {{x}}) e. ( Tfin A +o 1c))
24		pw1eq 4144	. . . . . . . . . . . 12 |- (a = (b u. {x}) -> ~P1 a = ~P1 (b u. {x}))
25		pw1un 4164	. . . . . . . . . . . . 13 |- ~P1 (b u. {x}) = (~P1 b u. ~P1 {x})
26	15	pw1sn 4166	. . . . . . . . . . . . . 14 |- ~P1 {x} = {{x}}
27	26	uneq2i 3416	. . . . . . . . . . . . 13 |- (~P1 b u. ~P1 {x}) = (~P1 b u. {{x}})
28	25, 27	eqtri 2373	. . . . . . . . . . . 12 |- ~P1 (b u. {x}) = (~P1 b u. {{x}})
29	24, 28	syl6eq 2401	. . . . . . . . . . 11 |- (a = (b u. {x}) -> ~P1 a = (~P1 b u. {{x}}))
30	29	eleq1d 2419	. . . . . . . . . 10 |- (a = (b u. {x}) -> (~P1 a e. ( Tfin A +o 1c) <-> (~P1 b u. {{x}}) e. ( Tfin A +o 1c)))
31	23, 30	syl5ibrcom 213	. . . . . . . . 9 |- ((A e. Nn /\ (b e. A /\ x e. ∼ b)) -> (a = (b u. {x}) -> ~P1 a e. ( Tfin A +o 1c)))
32	31	rexlimdvva 2746	. . . . . . . 8 |- (A e. Nn -> (E.b e. A E.x e. ∼ ba = (b u. {x}) -> ~P1 a e. ( Tfin A +o 1c)))
33	12, 32	syl5bi 208	. . . . . . 7 |- (A e. Nn -> (a e. (A +o 1c) -> ~P1 a e. ( Tfin A +o 1c)))
34	33	imp 418	. . . . . 6 |- ((A e. Nn /\ a e. (A +o 1c)) -> ~P1 a e. ( Tfin A +o 1c))
35		nnceleq 4431	. . . . . 6 |- ((( Tfin (A +o 1c) e. Nn /\ ( Tfin A +o 1c) e. Nn ) /\ (~P1 a e. Tfin (A +o 1c) /\ ~P1 a e. ( Tfin A +o 1c))) -> Tfin (A +o 1c) = ( Tfin A +o 1c))
36	5, 9, 11, 34, 35	syl22anc 1183	. . . . 5 |- ((A e. Nn /\ a e. (A +o 1c)) -> Tfin (A +o 1c) = ( Tfin A +o 1c))
37	36	ex 423	. . . 4 |- (A e. Nn -> (a e. (A +o 1c) -> Tfin (A +o 1c) = ( Tfin A +o 1c)))
38	37	exlimdv 1636	. . 3 |- (A e. Nn -> (E.a a e. (A +o 1c) -> Tfin (A +o 1c) = ( Tfin A +o 1c)))
39	1, 38	syl5bi 208	. 2 |- (A e. Nn -> ((A +o 1c) =/= (/) -> Tfin (A +o 1c) = ( Tfin A +o 1c)))
40	39	imp 418	1 |- ((A e. Nn /\ (A +o 1c) =/= (/)) -> Tfin (A +o 1c) = ( Tfin A +o 1c))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616   ∼ ccompl 3206   u. cun 3208  (/)c0 3551  {csn 3738  1_cc1c 4135  ~P₁ cpw1 4136   Nn cnnc 4374   +o cplc 4376   T_fin ctfin 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-tfin 4444

This theorem is referenced by:  tfin1c  4500  tfinltfinlem1  4501  sfintfin  4533  tfinnn  4535