Theorem ordunisuc2 3105

Description: An ordinal equal to its union contains the successor of each of its members.

Assertion

Ref	Expression
ordunisuc2	|- (Ord A -> (A = U.A <-> A.x e. A suc x e. A))

Distinct variable group:   x,A

Proof of Theorem ordunisuc2

Step	Hyp	Ref	Expression
1		orduninsuc 3104	. 2 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
2		ordtri3 2973	. . . . . . . . 9 |- ((Ord A /\ Ord suc x) -> (A = suc x <-> -. (A e. suc x \/ suc x e. A)))
3		suceloni 3052	. . . . . . . . . 10 |- (x e. On -> suc x e. On)
4		eloni 2948	. . . . . . . . . 10 |- (suc x e. On -> Ord suc x)
5	3, 4	syl 10	. . . . . . . . 9 |- (x e. On -> Ord suc x)
6	2, 5	sylan2 451	. . . . . . . 8 |- ((Ord A /\ x e. On) -> (A = suc x <-> -. (A e. suc x \/ suc x e. A)))
7	6	con2bid 524	. . . . . . 7 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) <-> -. A = suc x))
8		onnbtwn 3054	. . . . . . . . . . . . 13 |- (x e. On -> -. (x e. A /\ A e. suc x))
9		imnan 242	. . . . . . . . . . . . 13 |- ((x e. A -> -. A e. suc x) <-> -. (x e. A /\ A e. suc x))
10	8, 9	sylibr 200	. . . . . . . . . . . 12 |- (x e. On -> (x e. A -> -. A e. suc x))
11	10	con2d 91	. . . . . . . . . . 11 |- (x e. On -> (A e. suc x -> -. x e. A))
12		pm2.21 76	. . . . . . . . . . 11 |- (-. x e. A -> (x e. A -> suc x e. A))
13	11, 12	syl6 22	. . . . . . . . . 10 |- (x e. On -> (A e. suc x -> (x e. A -> suc x e. A)))
14	13	adantl 388	. . . . . . . . 9 |- ((Ord A /\ x e. On) -> (A e. suc x -> (x e. A -> suc x e. A)))
15		ax-1 4	. . . . . . . . . 10 |- (suc x e. A -> (x e. A -> suc x e. A))
16	15	a1i 8	. . . . . . . . 9 |- ((Ord A /\ x e. On) -> (suc x e. A -> (x e. A -> suc x e. A)))
17	14, 16	jaod 424	. . . . . . . 8 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) -> (x e. A -> suc x e. A)))
18		ordtri2or 3067	. . . . . . . . . . . . . 14 |- ((Ord x /\ Ord A) -> (x e. A \/ A (_ x))
19		eloni 2948	. . . . . . . . . . . . . 14 |- (x e. On -> Ord x)
20	18, 19	sylan 448	. . . . . . . . . . . . 13 |- ((x e. On /\ Ord A) -> (x e. A \/ A (_ x))
21	20	ancoms 436	. . . . . . . . . . . 12 |- ((Ord A /\ x e. On) -> (x e. A \/ A (_ x))
22		orcom 246	. . . . . . . . . . . 12 |- ((x e. A \/ A (_ x) <-> (A (_ x \/ x e. A))
23	21, 22	sylib 198	. . . . . . . . . . 11 |- ((Ord A /\ x e. On) -> (A (_ x \/ x e. A))
24	23	adantr 389	. . . . . . . . . 10 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A (_ x \/ x e. A))
25		ordsssuc2 3049	. . . . . . . . . . . . 13 |- ((Ord A /\ x e. On) -> (A (_ x <-> A e. suc x))
26	25	biimpd 153	. . . . . . . . . . . 12 |- ((Ord A /\ x e. On) -> (A (_ x -> A e. suc x))
27	26	adantr 389	. . . . . . . . . . 11 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A (_ x -> A e. suc x))
28		pm3.27 323	. . . . . . . . . . 11 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (x e. A -> suc x e. A))
29	27, 28	orim12d 563	. . . . . . . . . 10 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> ((A (_ x \/ x e. A) -> (A e. suc x \/ suc x e. A)))
30	24, 29	mpd 26	. . . . . . . . 9 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A e. suc x \/ suc x e. A))
31	30	ex 373	. . . . . . . 8 |- ((Ord A /\ x e. On) -> ((x e. A -> suc x e. A) -> (A e. suc x \/ suc x e. A)))
32	17, 31	impbid 514	. . . . . . 7 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) <-> (x e. A -> suc x e. A)))
33	7, 32	bitr3d 528	. . . . . 6 |- ((Ord A /\ x e. On) -> (-. A = suc x <-> (x e. A -> suc x e. A)))
34	33	pm5.74da 584	. . . . 5 |- (Ord A -> ((x e. On -> -. A = suc x) <-> (x e. On -> (x e. A -> suc x e. A))))
35		pm3.27 323	. . . . . . . 8 |- ((x e. On /\ x e. A) -> x e. A)
36		ordelon 2961	. . . . . . . . . 10 |- ((Ord A /\ x e. A) -> x e. On)
37	36	ex 373	. . . . . . . . 9 |- (Ord A -> (x e. A -> x e. On))
38	37	ancrd 299	. . . . . . . 8 |- (Ord A -> (x e. A -> (x e. On /\ x e. A)))
39	35, 38	impbid2 516	. . . . . . 7 |- (Ord A -> ((x e. On /\ x e. A) <-> x e. A))
40	39	imbi1d 611	. . . . . 6 |- (Ord A -> (((x e. On /\ x e. A) -> suc x e. A) <-> (x e. A -> suc x e. A)))
41		impexp 347	. . . . . 6 |- (((x e. On /\ x e. A) -> suc x e. A) <-> (x e. On -> (x e. A -> suc x e. A)))
42	40, 41	syl5bbr 532	. . . . 5 |- (Ord A -> ((x e. On -> (x e. A -> suc x e. A)) <-> (x e. A -> suc x e. A)))
43	34, 42	bitrd 526	. . . 4 |- (Ord A -> ((x e. On -> -. A = suc x) <-> (x e. A -> suc x e. A)))
44	43	ralbidv2 1657	. . 3 |- (Ord A -> (A.x e. On -. A = suc x <-> A.x e. A suc x e. A))
45		ralnex 1645	. . 3 |- (A.x e. On -. A = suc x <-> -. E.x e. On A = suc x)
46	44, 45	syl5bbr 532	. 2 |- (Ord A -> (-. E.x e. On A = suc x <-> A.x e. A suc x e. A))
47	1, 46	bitrd 526	1 |- (Ord A -> (A = U.A <-> A.x e. A suc x e. A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638   (_ wss 2037  U.cuni 2493  Ord word 2937  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  dflim4 3109

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

Copyright terms: Public domain