Theorem lealltlt2 16635

Description: Alternative definition for <_ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)

Assertion

Ref	Expression
lealltlt2	|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> A. x e. RR ( B < x -> A < x ) ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem lealltlt2

Step	Hyp	Ref	Expression
1		lelttr 8378	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ x e. RR ) -> ( ( A <_ B /\ B < x ) -> A < x ) )
2	1	expd 258	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ x e. RR ) -> ( A <_ B -> ( B < x -> A < x ) ) )
3	2	3expia 1232	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( x e. RR -> ( A <_ B -> ( B < x -> A < x ) ) ) )
4	3	com23 78	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> ( x e. RR -> ( B < x -> A < x ) ) ) )
5	4	ralrimdv 2623	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> A. x e. RR ( B < x -> A < x ) ) )
6		breq2 4118	. . . . . . 7 |- ( x = A -> ( B < x <-> B < A ) )
7		breq2 4118	. . . . . . 7 |- ( x = A -> ( A < x <-> A < A ) )
8	6, 7	imbi12d 234	. . . . . 6 |- ( x = A -> ( ( B < x -> A < x ) <-> ( B < A -> A < A ) ) )
9	8	rspcv 2919	. . . . 5 |- ( A e. RR -> ( A. x e. RR ( B < x -> A < x ) -> ( B < A -> A < A ) ) )
10		ltnr 8366	. . . . 5 |- ( A e. RR -> -. A < A )
11		con3 647	. . . . 5 |- ( ( B < A -> A < A ) -> ( -. A < A -> -. B < A ) )
12	9, 10, 11	syl6ci 1491	. . . 4 |- ( A e. RR -> ( A. x e. RR ( B < x -> A < x ) -> -. B < A ) )
13	12	adantr 276	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A. x e. RR ( B < x -> A < x ) -> -. B < A ) )
14		lenlt 8365	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
15	13, 14	sylibrd 169	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A. x e. RR ( B < x -> A < x ) -> A <_ B ) )
16	5, 15	impbid 129	1 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> A. x e. RR ( B < x -> A < x ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   class class class wbr 4114   RRcr 8142   < clt 8324   <_ cle 8325

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255  ax-pre-ltwlin 8256

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330

This theorem is referenced by: (None)