Theorem lealltlt2 16735

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 8408	. . . . . 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 1236	. . . 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 2629	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> A. x e. RR ( B < x -> A < x ) ) )
6		breq2 4132	. . . . . . 7 |- ( x = A -> ( B < x <-> B < A ) )
7		breq2 4132	. . . . . . 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 2925	. . . . 5 |- ( A e. RR -> ( A. x e. RR ( B < x -> A < x ) -> ( B < A -> A < A ) ) )
10		ltnr 8396	. . . . 5 |- ( A e. RR -> -. A < A )
11		con3 651	. . . . 5 |- ( ( B < A -> A < A ) -> ( -. A < A -> -. B < A ) )
12	9, 10, 11	syl6ci 1495	. . . 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 8395	. . 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 1009   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4128   RRcr 8172   < clt 8354   <_ cle 8355

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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-pre-ltirr 8285  ax-pre-ltwlin 8286

This theorem 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-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-cnv 4780  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360

This theorem is referenced by: (None)