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Theorem iunss 4007
 Description: Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss (x A B Cx A B C)
Distinct variable group:   x,C
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem iunss
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-iun 3971 . . 3 x A B = {y x A y B}
21sseq1i 3295 . 2 (x A B C ↔ {y x A y B} C)
3 abss 3335 . 2 ({y x A y B} Cy(x A y By C))
4 dfss2 3262 . . . 4 (B Cy(y By C))
54ralbii 2638 . . 3 (x A B Cx A y(y By C))
6 ralcom4 2877 . . 3 (x A y(y By C) ↔ yx A (y By C))
7 r19.23v 2730 . . . 4 (x A (y By C) ↔ (x A y By C))
87albii 1566 . . 3 (yx A (y By C) ↔ y(x A y By C))
95, 6, 83bitrri 263 . 2 (y(x A y By C) ↔ x A B C)
102, 3, 93bitri 262 1 (x A B Cx A B C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615   ⊆ 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-ss 3259  df-iun 3971 This theorem is referenced by:  iunss2  4011
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