Theorem intssuni2 2560

Description: Subclass relationship for intersection and union.

Assertion

Ref	Expression
intssuni2	|- ((A (_ B /\ A =/= (/)) -> |^|A (_ U.B)

Proof of Theorem intssuni2

Step	Hyp	Ref	Expression
1		intssuni 2559	. 2 |- (A =/= (/) -> |^|A (_ U.A)
2		uniss 2525	. 2 |- (A (_ B -> U.A (_ U.B)
3	1, 2	sylan9ssr 2079	1 |- ((A (_ B /\ A =/= (/)) -> |^|A (_ U.B)

Syntax hints:   -> wi 3   /\ wa 223   =/= wne 1588   (_ wss 2050  (/)c0 2283  U.cuni 2507  |^|cint 2537

This theorem is referenced by:  fiiu 10444  fiv 10470  fiiu2 10473

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-uni 2508  df-int 2538

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