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Theorem iunxsng 3706
 Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
Hypothesis
Ref Expression
iunxsng.1 (x = AB = 𝐶)
Assertion
Ref Expression
iunxsng (A 𝑉 x {A}B = 𝐶)
Distinct variable groups:   x,A   x,𝐶
Allowed substitution hints:   B(x)   𝑉(x)

Proof of Theorem iunxsng
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eliun 3635 . . 3 (y x {A}Bx {A}y B)
2 iunxsng.1 . . . . 5 (x = AB = 𝐶)
32eleq2d 2089 . . . 4 (x = A → (y By 𝐶))
43rexsng 3386 . . 3 (A 𝑉 → (x {A}y By 𝐶))
51, 4syl5bb 181 . 2 (A 𝑉 → (y x {A}By 𝐶))
65eqrdv 2020 1 (A 𝑉 x {A}B = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  {csn 3350  ∪ ciun 3631 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-sn 3356  df-iun 3633 This theorem is referenced by:  iunxsn  3707  rdgisuc1  5891  oasuc  5959  omsuc  5966
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