Theorem dedekindicclemub 15651

Description: Lemma for dedekindicc 15657. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)

Hypotheses

Ref	Expression
dedekindicc.a	|- ( ph -> A e. RR )
dedekindicc.b	|- ( ph -> B e. RR )
dedekindicc.lss	|- ( ph -> L C_ ( A [,] B ) )
dedekindicc.uss	|- ( ph -> U C_ ( A [,] B ) )
dedekindicc.lm	|- ( ph -> E. q e. ( A [,] B ) q e. L )
dedekindicc.um	|- ( ph -> E. r e. ( A [,] B ) r e. U )
dedekindicc.lr	|- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
dedekindicc.ur	|- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
dedekindicc.disj	|- ( ph -> ( L i^i U ) = (/) )
dedekindicc.loc	|- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )

Assertion

Ref	Expression
dedekindicclemub	|- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )

Distinct variable groups:   A, q, r, y   x, A, y   B, q, r, y   x, B   L, q, y   x, L   U, q, r, y   ph, q, y

Allowed substitution hints:   ph( x, r)   U( x)   L( r)

Proof of Theorem dedekindicclemub

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindicc.um	. . 3 |- ( ph -> E. r e. ( A [,] B ) r e. U )
2		eleq1w 2299	. . . 4 |- ( r = a -> ( r e. U <-> a e. U ) )
3	2	cbvrexv 2787	. . 3 |- ( E. r e. ( A [,] B ) r e. U <-> E. a e. ( A [,] B ) a e. U )
4	1, 3	sylib 122	. 2 |- ( ph -> E. a e. ( A [,] B ) a e. U )
5		simprl 535	. . 3 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> a e. ( A [,] B ) )
6		dedekindicc.a	. . . . 5 |- ( ph -> A e. RR )
7	6	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A e. RR )
8		dedekindicc.b	. . . . 5 |- ( ph -> B e. RR )
9	8	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> B e. RR )
10		dedekindicc.lss	. . . . 5 |- ( ph -> L C_ ( A [,] B ) )
11	10	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> L C_ ( A [,] B ) )
12		dedekindicc.uss	. . . . 5 |- ( ph -> U C_ ( A [,] B ) )
13	12	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> U C_ ( A [,] B ) )
14		dedekindicc.lm	. . . . 5 |- ( ph -> E. q e. ( A [,] B ) q e. L )
15	14	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> E. q e. ( A [,] B ) q e. L )
16	1	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> E. r e. ( A [,] B ) r e. U )
17		dedekindicc.lr	. . . . 5 |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
18	17	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
19		dedekindicc.ur	. . . . 5 |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
20	19	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )
21		dedekindicc.disj	. . . . 5 |- ( ph -> ( L i^i U ) = (/) )
22	21	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> ( L i^i U ) = (/) )
23		dedekindicc.loc	. . . . 5 |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )
24	23	adantr 276	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )
25		simprr 537	. . . 4 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> a e. U )
26	7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25	dedekindicclemuub 15650	. . 3 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> A. y e. L y < a )
27		brralrspcev 4184	. . 3 |- ( ( a e. ( A [,] B ) /\ A. y e. L y < a ) -> E. x e. ( A [,] B ) A. y e. L y < x )
28	5, 26, 27	syl2anc 415	. 2 |- ( ( ph /\ ( a e. ( A [,] B ) /\ a e. U ) ) -> E. x e. ( A [,] B ) A. y e. L y < x )
29	4, 28	rexlimddv 2673	1 |- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   i^i cin 3219   C_ wss 3220   (/)c0 3520   class class class wbr 4125  (class class class)co 6075   RRcr 8168   < clt 8350   [,]cicc 10272

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 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-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283

This theorem depends on definitions:  df-bi 117  df-3or 1010  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-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-icc 10276

This theorem is referenced by: (None)