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Theorem unisngl 20479
 Description: Taking the union of the set of singletons recovers the initial set. (Contributed by Thierry Arnoux, 9-Jan-2020.)
Hypothesis
Ref Expression
dissnref.c
Assertion
Ref Expression
unisngl
Distinct variable groups:   ,,   ,,

Proof of Theorem unisngl
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dissnref.c . . 3
21unieqi 4166 . 2
3 simpl 458 . . . . . . . . 9
4 simpr 462 . . . . . . . . 9
53, 4eleqtrd 2503 . . . . . . . 8
65exlimiv 1770 . . . . . . 7
7 eqid 2423 . . . . . . . 8
8 snex 4600 . . . . . . . . 9
9 eleq2 2490 . . . . . . . . . 10
10 eqeq1 2427 . . . . . . . . . 10
119, 10anbi12d 715 . . . . . . . . 9
128, 11spcev 3111 . . . . . . . 8
137, 12mpan2 675 . . . . . . 7
146, 13impbii 190 . . . . . 6
15 elsn 3950 . . . . . 6
16 equcom 1848 . . . . . 6
1714, 15, 163bitri 274 . . . . 5
1817rexbii 2861 . . . 4
19 r19.42v 2917 . . . . . 6
2019exbii 1712 . . . . 5
21 rexcom4 3038 . . . . 5
22 eluniab 4168 . . . . 5
2320, 21, 223bitr4ri 281 . . . 4
24 risset 2887 . . . 4
2518, 23, 243bitr4i 280 . . 3
2625eqriv 2420 . 2
272, 26eqtr2i 2446 1
 Colors of variables: wff setvar class Syntax hints:   wa 370   wceq 1437  wex 1657   wcel 1872  cab 2409  wrex 2710  csn 3936  cuni 4157 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pr 4598 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-v 3019  df-dif 3377  df-un 3379  df-nul 3700  df-sn 3937  df-pr 3939  df-uni 4158 This theorem is referenced by:  dissnref  20480  dissnlocfin  20481
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