Theorem oawordriexmid 6637

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 6636. (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 6588	. . . . 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 1229	. . . . . . 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 2585	. . . . 5 |- A. c e. On ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )
6		oveq2 6025	. . . . . . . . 9 |- ( c = 1o -> ( a +o c ) = ( a +o 1o ) )
7		oveq2 6025	. . . . . . . . 9 |- ( c = 1o -> ( b +o c ) = ( b +o 1o ) )
8	6, 7	sseq12d 3258	. . . . . . . 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 2906	. . . . 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 6634	. . . . . 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 6634	. . . . . 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 3258	. . . 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 2586	. 2 |- A. a e. On A. b e. On ( a C_ b -> suc a C_ suc b )
20	19	onsucsssucexmid 4625	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   Oncon0 4460   suc csuc 4462  (class class class)co 6017   1oc1o 6574   +o coa 6578

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-oadd 6585

This theorem is referenced by: (None)