Theorem nulsslt 31908

Description: The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
nulsslt	|- ( A e. ~P No -> (/) < <s A )

Proof of Theorem nulsslt

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( A e. ~P No -> A e. _V )
2		0ex 4790	. . 3 |- (/) e. _V
3	1, 2	jctil 560	. 2 |- ( A e. ~P No -> ( (/) e. _V /\ A e. _V ) )
4		0ss 3972	. . . 4 |- (/) C_ No
5	4	a1i 11	. . 3 |- ( A e. ~P No -> (/) C_ No )
6		elpwi 4168	. . 3 |- ( A e. ~P No -> A C_ No )
7		ral0 4076	. . . 4 |- A. x e. (/) A. y e. A x <s y
8	7	a1i 11	. . 3 |- ( A e. ~P No -> A. x e. (/) A. y e. A x <s y )
9	5, 6, 8	3jca 1242	. 2 |- ( A e. ~P No -> ( (/) C_ No /\ A C_ No /\ A. x e. (/) A. y e. A x <s y ) )
10		brsslt 31900	. 2 |- ( (/) < <s A <-> ( ( (/) e. _V /\ A e. _V ) /\ ( (/) C_ No /\ A C_ No /\ A. x e. (/) A. y e. A x <s y ) ) )
11	3, 9, 10	sylanbrc 698	1 |- ( A e. ~P No -> (/) < <s A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   Nocsur 31793   <scslt 31794   < <scsslt 31896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-sslt 31897

This theorem is referenced by: (None)