Theorem nulssgt 31909

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

Assertion

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

Proof of Theorem nulssgt

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	jctir 561	. 2 |- ( A e. ~P No -> ( A e. _V /\ (/) e. _V ) )
4		elpwi 4168	. . 3 |- ( A e. ~P No -> A C_ No )
5		0ss 3972	. . . 4 |- (/) C_ No
6	5	a1i 11	. . 3 |- ( A e. ~P No -> (/) C_ No )
7		ral0 4076	. . . . 5 |- A. y e. (/) x <s y
8	7	rgenw 2924	. . . 4 |- A. x e. A A. y e. (/) x <s y
9	8	a1i 11	. . 3 |- ( A e. ~P No -> A. x e. A A. y e. (/) x <s y )
10	4, 6, 9	3jca 1242	. 2 |- ( A e. ~P No -> ( A C_ No /\ (/) C_ No /\ A. x e. A A. y e. (/) x <s y ) )
11		brsslt 31900	. 2 |- ( A < <s (/) <-> ( ( A e. _V /\ (/) e. _V ) /\ ( A C_ No /\ (/) C_ No /\ A. x e. A A. y e. (/) x <s y ) ) )
12	3, 10, 11	sylanbrc 698	1 |- ( A e. ~P No -> A < <s (/) )

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)