Theorem oawordriexmid 6438

Description: A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6437. (Contributed by Jim Kingdon, 15-May-2022.)

Hypothesis

Ref	Expression
oawordriexmid.1	|- ( ( a e. On /\ b e. On /\ c e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )

Assertion

Ref	Expression
oawordriexmid	|- ( ph \/ -. ph )

Distinct variable groups:   a, b, c   ph, a

Allowed substitution hints:   ph( b, c)

Proof of Theorem oawordriexmid

Step	Hyp	Ref	Expression
1		1on 6391	. . . . 5 |- 1o e. On
2		oawordriexmid.1	. . . . . . . 8 |- ( ( a e. On /\ b e. On /\ c e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )
3	2	3expa 1193	. . . . . . 7 |- ( ( ( a e. On /\ b e. On ) /\ c e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )
4	3	expcom 115	. . . . . 6 |- ( c e. On -> ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) ) )
5	4	rgen 2519	. . . . 5 |- A. c e. On ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )
6		oveq2 5850	. . . . . . . . 9 |- ( c = 1o -> ( a +o c ) = ( a +o 1o ) )
7		oveq2 5850	. . . . . . . . 9 |- ( c = 1o -> ( b +o c ) = ( b +o 1o ) )
8	6, 7	sseq12d 3173	. . . . . . . 8 |- ( c = 1o -> ( ( a +o c ) C_ ( b +o c ) <-> ( a +o 1o ) C_ ( b +o 1o ) ) )
9	8	imbi2d 229	. . . . . . 7 |- ( c = 1o -> ( ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) <-> ( a C_ b -> ( a +o 1o ) C_ ( b +o 1o ) ) ) )
10	9	imbi2d 229	. . . . . 6 |- ( c = 1o -> ( ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) ) <-> ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o 1o ) C_ ( b +o 1o ) ) ) ) )
11	10	rspcv 2826	. . . . 5 |- ( 1o e. On -> ( A. c e. On ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) ) -> ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o 1o ) C_ ( b +o 1o ) ) ) ) )
12	1, 5, 11	mp2 16	. . . 4 |- ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o 1o ) C_ ( b +o 1o ) ) )
13		oa1suc 6435	. . . . . 6 |- ( a e. On -> ( a +o 1o ) = suc a )
14	13	adantr 274	. . . . 5 |- ( ( a e. On /\ b e. On ) -> ( a +o 1o ) = suc a )
15		oa1suc 6435	. . . . . 6 |- ( b e. On -> ( b +o 1o ) = suc b )
16	15	adantl 275	. . . . 5 |- ( ( a e. On /\ b e. On ) -> ( b +o 1o ) = suc b )
17	14, 16	sseq12d 3173	. . . 4 |- ( ( a e. On /\ b e. On ) -> ( ( a +o 1o ) C_ ( b +o 1o ) <-> suc a C_ suc b ) )
18	12, 17	sylibd 148	. . 3 |- ( ( a e. On /\ b e. On ) -> ( a C_ b -> suc a C_ suc b ) )
19	18	rgen2a 2520	. 2 |- A. a e. On A. b e. On ( a C_ b -> suc a C_ suc b )
20	19	onsucsssucexmid 4504	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   Oncon0 4341   suc csuc 4343  (class class class)co 5842   1oc1o 6377   +o coa 6381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388

This theorem is referenced by: (None)