Theorem oawordriexmid 6474

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 6473. (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 6427	. . . . 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 1203	. . . . . . 7 |- ( ( ( a e. On /\ b e. On ) /\ c e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )
4	3	expcom 116	. . . . . 6 |- ( c e. On -> ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) ) )
5	4	rgen 2530	. . . . 5 |- A. c e. On ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )
6		oveq2 5886	. . . . . . . . 9 |- ( c = 1o -> ( a +o c ) = ( a +o 1o ) )
7		oveq2 5886	. . . . . . . . 9 |- ( c = 1o -> ( b +o c ) = ( b +o 1o ) )
8	6, 7	sseq12d 3188	. . . . . . . 8 |- ( c = 1o -> ( ( a +o c ) C_ ( b +o c ) <-> ( a +o 1o ) C_ ( b +o 1o ) ) )
9	8	imbi2d 230	. . . . . . 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 230	. . . . . 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 2839	. . . . 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 6471	. . . . . 6 |- ( a e. On -> ( a +o 1o ) = suc a )
14	13	adantr 276	. . . . 5 |- ( ( a e. On /\ b e. On ) -> ( a +o 1o ) = suc a )
15		oa1suc 6471	. . . . . 6 |- ( b e. On -> ( b +o 1o ) = suc b )
16	15	adantl 277	. . . . 5 |- ( ( a e. On /\ b e. On ) -> ( b +o 1o ) = suc b )
17	14, 16	sseq12d 3188	. . . 4 |- ( ( a e. On /\ b e. On ) -> ( ( a +o 1o ) C_ ( b +o 1o ) <-> suc a C_ suc b ) )
18	12, 17	sylibd 149	. . 3 |- ( ( a e. On /\ b e. On ) -> ( a C_ b -> suc a C_ suc b ) )
19	18	rgen2a 2531	. 2 |- A. a e. On A. b e. On ( a C_ b -> suc a C_ suc b )
20	19	onsucsssucexmid 4528	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   C_ wss 3131   Oncon0 4365   suc csuc 4367  (class class class)co 5878   1oc1o 6413   +o coa 6417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-oadd 6424

This theorem is referenced by: (None)