Theorem ssuniOLD 4460

Description: Obsolete proof of ssuni 4459 as of 26-Jul-2021. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ssuniOLD	|- ( ( A C_ B /\ B e. C ) -> A C_ U. C )

Proof of Theorem ssuniOLD

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

Step	Hyp	Ref	Expression
1		eleq2 2690	. . . . . . 7 |- ( x = B -> ( y e. x <-> y e. B ) )
2	1	imbi1d 331	. . . . . 6 |- ( x = B -> ( ( y e. x -> y e. U. C ) <-> ( y e. B -> y e. U. C ) ) )
3		elunii 4441	. . . . . . 7 |- ( ( y e. x /\ x e. C ) -> y e. U. C )
4	3	expcom 451	. . . . . 6 |- ( x e. C -> ( y e. x -> y e. U. C ) )
5	2, 4	vtoclga 3272	. . . . 5 |- ( B e. C -> ( y e. B -> y e. U. C ) )
6	5	imim2d 57	. . . 4 |- ( B e. C -> ( ( y e. A -> y e. B ) -> ( y e. A -> y e. U. C ) ) )
7	6	alimdv 1845	. . 3 |- ( B e. C -> ( A. y ( y e. A -> y e. B ) -> A. y ( y e. A -> y e. U. C ) ) )
8		dfss2 3591	. . 3 |- ( A C_ B <-> A. y ( y e. A -> y e. B ) )
9		dfss2 3591	. . 3 |- ( A C_ U. C <-> A. y ( y e. A -> y e. U. C ) )
10	7, 8, 9	3imtr4g 285	. 2 |- ( B e. C -> ( A C_ B -> A C_ U. C ) )
11	10	impcom 446	1 |- ( ( A C_ B /\ B e. C ) -> A C_ U. C )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437

This theorem is referenced by: (None)