New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iunxdif2 GIF version

Theorem iunxdif2 4014
 Description: Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
Hypothesis
Ref Expression
iunxdif2.1 (x = yC = D)
Assertion
Ref Expression
iunxdif2 (x A y (A B)C Dy (A B)D = x A C)
Distinct variable groups:   x,y,A   x,B,y   y,C   x,D
Allowed substitution hints:   C(x)   D(y)

Proof of Theorem iunxdif2
StepHypRef Expression
1 iunss2 4011 . . 3 (x A y (A B)C Dx A C y (A B)D)
2 difss 3393 . . . . 5 (A B) A
3 iunss1 3980 . . . . 5 ((A B) Ay (A B)D y A D)
42, 3ax-mp 8 . . . 4 y (A B)D y A D
5 iunxdif2.1 . . . . 5 (x = yC = D)
65cbviunv 4005 . . . 4 x A C = y A D
74, 6sseqtr4i 3304 . . 3 y (A B)D x A C
81, 7jctil 523 . 2 (x A y (A B)C D → (y (A B)D x A C x A C y (A B)D))
9 eqss 3287 . 2 (y (A B)D = x A C ↔ (y (A B)D x A C x A C y (A B)D))
108, 9sylibr 203 1 (x A y (A B)C Dy (A B)D = x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ∀wral 2614  ∃wrex 2615   ∖ cdif 3206   ⊆ wss 3257  ∪ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-iun 3971 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator