Theorem oawordriexmid 6747

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 6746. (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 6698	. . . . 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 1234	. . . . . . 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 2603	. . . . 5 |- A. c e. On ( ( a e. On /\ b e. On ) -> ( a C_ b -> ( a +o c ) C_ ( b +o c ) ) )
6		oveq2 6096	. . . . . . . . 9 |- ( c = 1o -> ( a +o c ) = ( a +o 1o ) )
7		oveq2 6096	. . . . . . . . 9 |- ( c = 1o -> ( b +o c ) = ( b +o 1o ) )
8	6, 7	sseq12d 3279	. . . . . . . 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 2925	. . . . 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 6744	. . . . . 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 6744	. . . . . 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 3279	. . . 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 2604	. 2 |- A. a e. On A. b e. On ( a C_ b -> suc a C_ suc b )
20	19	onsucsssucexmid 4676	1 |- ( ph \/ -. ph )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   Oncon0 4510   suc csuc 4512  (class class class)co 6088   1oc1o 6684   +o coa 6688

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695

This theorem is used by: (None)