Metamath Proof Explorer


Theorem oaordi

Description: Ordering property of ordinal addition. Proposition 8.4 of TakeutiZaring p. 58. (Contributed by NM, 5-Dec-2004)

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

Proof

Step Hyp Ref Expression
1 onelon
 |-  ( ( B e. On /\ A e. B ) -> A e. On )
2 1 adantll
 |-  ( ( ( C e. On /\ B e. On ) /\ A e. B ) -> A e. On )
3 eloni
 |-  ( B e. On -> Ord B )
4 ordsucss
 |-  ( Ord B -> ( A e. B -> suc A C_ B ) )
5 3 4 syl
 |-  ( B e. On -> ( A e. B -> suc A C_ B ) )
6 5 ad2antlr
 |-  ( ( ( C e. On /\ B e. On ) /\ A e. On ) -> ( A e. B -> suc A C_ B ) )
7 sucelon
 |-  ( A e. On <-> suc A e. On )
8 oveq2
 |-  ( x = suc A -> ( C +o x ) = ( C +o suc A ) )
9 8 sseq2d
 |-  ( x = suc A -> ( ( C +o suc A ) C_ ( C +o x ) <-> ( C +o suc A ) C_ ( C +o suc A ) ) )
10 9 imbi2d
 |-  ( x = suc A -> ( ( C e. On -> ( C +o suc A ) C_ ( C +o x ) ) <-> ( C e. On -> ( C +o suc A ) C_ ( C +o suc A ) ) ) )
11 oveq2
 |-  ( x = y -> ( C +o x ) = ( C +o y ) )
12 11 sseq2d
 |-  ( x = y -> ( ( C +o suc A ) C_ ( C +o x ) <-> ( C +o suc A ) C_ ( C +o y ) ) )
13 12 imbi2d
 |-  ( x = y -> ( ( C e. On -> ( C +o suc A ) C_ ( C +o x ) ) <-> ( C e. On -> ( C +o suc A ) C_ ( C +o y ) ) ) )
14 oveq2
 |-  ( x = suc y -> ( C +o x ) = ( C +o suc y ) )
15 14 sseq2d
 |-  ( x = suc y -> ( ( C +o suc A ) C_ ( C +o x ) <-> ( C +o suc A ) C_ ( C +o suc y ) ) )
16 15 imbi2d
 |-  ( x = suc y -> ( ( C e. On -> ( C +o suc A ) C_ ( C +o x ) ) <-> ( C e. On -> ( C +o suc A ) C_ ( C +o suc y ) ) ) )
17 oveq2
 |-  ( x = B -> ( C +o x ) = ( C +o B ) )
18 17 sseq2d
 |-  ( x = B -> ( ( C +o suc A ) C_ ( C +o x ) <-> ( C +o suc A ) C_ ( C +o B ) ) )
19 18 imbi2d
 |-  ( x = B -> ( ( C e. On -> ( C +o suc A ) C_ ( C +o x ) ) <-> ( C e. On -> ( C +o suc A ) C_ ( C +o B ) ) ) )
20 ssid
 |-  ( C +o suc A ) C_ ( C +o suc A )
21 20 2a1i
 |-  ( suc A e. On -> ( C e. On -> ( C +o suc A ) C_ ( C +o suc A ) ) )
22 sssucid
 |-  ( C +o y ) C_ suc ( C +o y )
23 sstr2
 |-  ( ( C +o suc A ) C_ ( C +o y ) -> ( ( C +o y ) C_ suc ( C +o y ) -> ( C +o suc A ) C_ suc ( C +o y ) ) )
24 22 23 mpi
 |-  ( ( C +o suc A ) C_ ( C +o y ) -> ( C +o suc A ) C_ suc ( C +o y ) )
25 oasuc
 |-  ( ( C e. On /\ y e. On ) -> ( C +o suc y ) = suc ( C +o y ) )
26 25 ancoms
 |-  ( ( y e. On /\ C e. On ) -> ( C +o suc y ) = suc ( C +o y ) )
27 26 sseq2d
 |-  ( ( y e. On /\ C e. On ) -> ( ( C +o suc A ) C_ ( C +o suc y ) <-> ( C +o suc A ) C_ suc ( C +o y ) ) )
28 24 27 syl5ibr
 |-  ( ( y e. On /\ C e. On ) -> ( ( C +o suc A ) C_ ( C +o y ) -> ( C +o suc A ) C_ ( C +o suc y ) ) )
29 28 ex
 |-  ( y e. On -> ( C e. On -> ( ( C +o suc A ) C_ ( C +o y ) -> ( C +o suc A ) C_ ( C +o suc y ) ) ) )
30 29 ad2antrr
 |-  ( ( ( y e. On /\ suc A e. On ) /\ suc A C_ y ) -> ( C e. On -> ( ( C +o suc A ) C_ ( C +o y ) -> ( C +o suc A ) C_ ( C +o suc y ) ) ) )
31 30 a2d
 |-  ( ( ( y e. On /\ suc A e. On ) /\ suc A C_ y ) -> ( ( C e. On -> ( C +o suc A ) C_ ( C +o y ) ) -> ( C e. On -> ( C +o suc A ) C_ ( C +o suc y ) ) ) )
32 sucssel
 |-  ( A e. On -> ( suc A C_ x -> A e. x ) )
33 7 32 sylbir
 |-  ( suc A e. On -> ( suc A C_ x -> A e. x ) )
34 limsuc
 |-  ( Lim x -> ( A e. x <-> suc A e. x ) )
35 34 biimpd
 |-  ( Lim x -> ( A e. x -> suc A e. x ) )
36 33 35 sylan9r
 |-  ( ( Lim x /\ suc A e. On ) -> ( suc A C_ x -> suc A e. x ) )
37 36 imp
 |-  ( ( ( Lim x /\ suc A e. On ) /\ suc A C_ x ) -> suc A e. x )
38 oveq2
 |-  ( y = suc A -> ( C +o y ) = ( C +o suc A ) )
39 38 ssiun2s
 |-  ( suc A e. x -> ( C +o suc A ) C_ U_ y e. x ( C +o y ) )
40 37 39 syl
 |-  ( ( ( Lim x /\ suc A e. On ) /\ suc A C_ x ) -> ( C +o suc A ) C_ U_ y e. x ( C +o y ) )
41 40 adantr
 |-  ( ( ( ( Lim x /\ suc A e. On ) /\ suc A C_ x ) /\ C e. On ) -> ( C +o suc A ) C_ U_ y e. x ( C +o y ) )
42 vex
 |-  x e. _V
43 oalim
 |-  ( ( C e. On /\ ( x e. _V /\ Lim x ) ) -> ( C +o x ) = U_ y e. x ( C +o y ) )
44 42 43 mpanr1
 |-  ( ( C e. On /\ Lim x ) -> ( C +o x ) = U_ y e. x ( C +o y ) )
45 44 ancoms
 |-  ( ( Lim x /\ C e. On ) -> ( C +o x ) = U_ y e. x ( C +o y ) )
46 45 adantlr
 |-  ( ( ( Lim x /\ suc A e. On ) /\ C e. On ) -> ( C +o x ) = U_ y e. x ( C +o y ) )
47 46 adantlr
 |-  ( ( ( ( Lim x /\ suc A e. On ) /\ suc A C_ x ) /\ C e. On ) -> ( C +o x ) = U_ y e. x ( C +o y ) )
48 41 47 sseqtrrd
 |-  ( ( ( ( Lim x /\ suc A e. On ) /\ suc A C_ x ) /\ C e. On ) -> ( C +o suc A ) C_ ( C +o x ) )
49 48 ex
 |-  ( ( ( Lim x /\ suc A e. On ) /\ suc A C_ x ) -> ( C e. On -> ( C +o suc A ) C_ ( C +o x ) ) )
50 49 a1d
 |-  ( ( ( Lim x /\ suc A e. On ) /\ suc A C_ x ) -> ( A. y e. x ( suc A C_ y -> ( C e. On -> ( C +o suc A ) C_ ( C +o y ) ) ) -> ( C e. On -> ( C +o suc A ) C_ ( C +o x ) ) ) )
51 10 13 16 19 21 31 50 tfindsg
 |-  ( ( ( B e. On /\ suc A e. On ) /\ suc A C_ B ) -> ( C e. On -> ( C +o suc A ) C_ ( C +o B ) ) )
52 51 exp31
 |-  ( B e. On -> ( suc A e. On -> ( suc A C_ B -> ( C e. On -> ( C +o suc A ) C_ ( C +o B ) ) ) ) )
53 7 52 syl5bi
 |-  ( B e. On -> ( A e. On -> ( suc A C_ B -> ( C e. On -> ( C +o suc A ) C_ ( C +o B ) ) ) ) )
54 53 com4r
 |-  ( C e. On -> ( B e. On -> ( A e. On -> ( suc A C_ B -> ( C +o suc A ) C_ ( C +o B ) ) ) ) )
55 54 imp31
 |-  ( ( ( C e. On /\ B e. On ) /\ A e. On ) -> ( suc A C_ B -> ( C +o suc A ) C_ ( C +o B ) ) )
56 oasuc
 |-  ( ( C e. On /\ A e. On ) -> ( C +o suc A ) = suc ( C +o A ) )
57 56 sseq1d
 |-  ( ( C e. On /\ A e. On ) -> ( ( C +o suc A ) C_ ( C +o B ) <-> suc ( C +o A ) C_ ( C +o B ) ) )
58 ovex
 |-  ( C +o A ) e. _V
59 sucssel
 |-  ( ( C +o A ) e. _V -> ( suc ( C +o A ) C_ ( C +o B ) -> ( C +o A ) e. ( C +o B ) ) )
60 58 59 ax-mp
 |-  ( suc ( C +o A ) C_ ( C +o B ) -> ( C +o A ) e. ( C +o B ) )
61 57 60 syl6bi
 |-  ( ( C e. On /\ A e. On ) -> ( ( C +o suc A ) C_ ( C +o B ) -> ( C +o A ) e. ( C +o B ) ) )
62 61 adantlr
 |-  ( ( ( C e. On /\ B e. On ) /\ A e. On ) -> ( ( C +o suc A ) C_ ( C +o B ) -> ( C +o A ) e. ( C +o B ) ) )
63 6 55 62 3syld
 |-  ( ( ( C e. On /\ B e. On ) /\ A e. On ) -> ( A e. B -> ( C +o A ) e. ( C +o B ) ) )
64 63 imp
 |-  ( ( ( ( C e. On /\ B e. On ) /\ A e. On ) /\ A e. B ) -> ( C +o A ) e. ( C +o B ) )
65 64 an32s
 |-  ( ( ( ( C e. On /\ B e. On ) /\ A e. B ) /\ A e. On ) -> ( C +o A ) e. ( C +o B ) )
66 2 65 mpdan
 |-  ( ( ( C e. On /\ B e. On ) /\ A e. B ) -> ( C +o A ) e. ( C +o B ) )
67 66 ex
 |-  ( ( C e. On /\ B e. On ) -> ( A e. B -> ( C +o A ) e. ( C +o B ) ) )
68 67 ancoms
 |-  ( ( B e. On /\ C e. On ) -> ( A e. B -> ( C +o A ) e. ( C +o B ) ) )