Theorem celsor 14443

Description: If all the elements of a set A are ordinal numbers and are parts of the set then A is an ordinal number.

Assertion

Ref	Expression
celsor	|- ((A e. B /\ A.x e. A (x e. On /\ x C_ A)) -> A e. On)

Distinct variable group:   x,A

Proof of Theorem celsor

Step	Hyp	Ref	Expression
1		r19.26 2219	. . . 4 |- (A.x e. A (x e. On /\ x C_ A) <-> (A.x e. A x e. On /\ A.x e. A x C_ A))
2		dftr3 3415	. . . . . . 7 |- (Tr A <-> A.x e. A x C_ A)
3	2	biimpri 169	. . . . . 6 |- (A.x e. A x C_ A -> Tr A)
4		eleq1 1957	. . . . . . . . 9 |- (x = y -> (x e. On <-> y e. On))
5	4	cbvralv 2280	. . . . . . . 8 |- (A.x e. A x e. On <-> A.y e. A y e. On)
6		raaanv 2977	. . . . . . . . . 10 |- (A.x e. A A.y e. A (x e. On /\ y e. On) <-> (A.x e. A x e. On /\ A.y e. A y e. On))
7		ordtri3or 3691	. . . . . . . . . . . . 13 |- ((Ord x /\ Ord y) -> (x e. y \/ x = y \/ y e. x))
8		eloni 3667	. . . . . . . . . . . . 13 |- (x e. On -> Ord x)
9		eloni 3667	. . . . . . . . . . . . 13 |- (y e. On -> Ord y)
10	7, 8, 9	syl2an 503	. . . . . . . . . . . 12 |- ((x e. On /\ y e. On) -> (x e. y \/ x = y \/ y e. x))
11	10	ralimi 2168	. . . . . . . . . . 11 |- (A.y e. A (x e. On /\ y e. On) -> A.y e. A (x e. y \/ x = y \/ y e. x))
12	11	ralimi 2168	. . . . . . . . . 10 |- (A.x e. A A.y e. A (x e. On /\ y e. On) -> A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x))
13	6, 12	sylbir 218	. . . . . . . . 9 |- ((A.x e. A x e. On /\ A.y e. A y e. On) -> A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x))
14	13	expcom 403	. . . . . . . 8 |- (A.y e. A y e. On -> (A.x e. A x e. On -> A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x)))
15	5, 14	sylbi 216	. . . . . . 7 |- (A.x e. A x e. On -> (A.x e. A x e. On -> A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x)))
16	15	pm2.43i 78	. . . . . 6 |- (A.x e. A x e. On -> A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x))
17	3, 16	anim12i 360	. . . . 5 |- ((A.x e. A x C_ A /\ A.x e. A x e. On) -> (Tr A /\ A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x)))
18	17	ancoms 484	. . . 4 |- ((A.x e. A x e. On /\ A.x e. A x C_ A) -> (Tr A /\ A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x)))
19	1, 18	sylbi 216	. . 3 |- (A.x e. A (x e. On /\ x C_ A) -> (Tr A /\ A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x)))
20	19	adantl 424	. 2 |- ((A e. B /\ A.x e. A (x e. On /\ x C_ A)) -> (Tr A /\ A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x)))
21		elong 3665	. . . 4 |- (A e. B -> (A e. On <-> Ord A))
22	21	adantr 425	. . 3 |- ((A e. B /\ A.x e. A (x e. On /\ x C_ A)) -> (A e. On <-> Ord A))
23		dford2 5711	. . 3 |- (Ord A <-> (Tr A /\ A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x)))
24	22, 23	syl6bb 595	. 2 |- ((A e. B /\ A.x e. A (x e. On /\ x C_ A)) -> (A e. On <-> (Tr A /\ A.x e. A A.y e. A (x e. y \/ x = y \/ y e. x))))
25	20, 24	mpbird 213	1 |- ((A e. B /\ A.x e. A (x e. On /\ x C_ A)) -> A e. On)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  Tr wtr 3411  Ord word 3656  Oncon0 3657

This theorem is referenced by:  inttaror 15277

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

Copyright terms: Public domain