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Theorem iunid 4021
 Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
Assertion
Ref Expression
iunid x A {x} = A
Distinct variable group:   x,A

Proof of Theorem iunid
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-sn 3741 . . . . 5 {x} = {y y = x}
2 equcom 1680 . . . . . 6 (y = xx = y)
32abbii 2465 . . . . 5 {y y = x} = {y x = y}
41, 3eqtri 2373 . . . 4 {x} = {y x = y}
54a1i 10 . . 3 (x A → {x} = {y x = y})
65iuneq2i 3987 . 2 x A {x} = x A {y x = y}
7 iunab 4012 . . 3 x A {y x = y} = {y x A x = y}
8 risset 2661 . . . 4 (y Ax A x = y)
98abbii 2465 . . 3 {y y A} = {y x A x = y}
10 abid2 2470 . . 3 {y y A} = A
117, 9, 103eqtr2i 2379 . 2 x A {y x = y} = A
126, 11eqtri 2373 1 x A {x} = A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  {csn 3737  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-sn 3741  df-iun 3971 This theorem is referenced by:  iunxpconst  4819  uniqs  5984
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