Theorem oawordriexmid 6616

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 6615. (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 6569	. . . . 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 1227	. . . . . . 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 2583	. . . . 5 |- A. c e. On ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )
6		oveq2 6009	. . . . . . . . 9 |- ( c = 1o -> ( a +o c ) = ( a +o 1o ) )
7		oveq2 6009	. . . . . . . . 9 |- ( c = 1o -> ( b +o c ) = ( b +o 1o ) )
8	6, 7	sseq12d 3255	. . . . . . . 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 2903	. . . . 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 6613	. . . . . 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 6613	. . . . . 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 3255	. . . 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 2584	. 2 |- A. a e. On A. b e. On ( a C_ b -> suc a C_ suc b )
20	19	onsucsssucexmid 4619	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   Oncon0 4454   suc csuc 4456  (class class class)co 6001   1oc1o 6555   +o coa 6559

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-oadd 6566

This theorem is referenced by: (None)