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Theorem snnex 4713
 Description: The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)
Assertion
Ref Expression
snnex
Distinct variable group:   ,

Proof of Theorem snnex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vprc 4341 . . . 4
2 vex 2959 . . . . . . . . . 10
32snid 3841 . . . . . . . . 9
4 a9ev 1668 . . . . . . . . . 10
5 sneq 3825 . . . . . . . . . . 11
65equcoms 1693 . . . . . . . . . 10
74, 6eximii 1587 . . . . . . . . 9
8 snex 4405 . . . . . . . . . 10
9 eleq2 2497 . . . . . . . . . . 11
10 eqeq1 2442 . . . . . . . . . . . 12
1110exbidv 1636 . . . . . . . . . . 11
129, 11anbi12d 692 . . . . . . . . . 10
138, 12spcev 3043 . . . . . . . . 9
143, 7, 13mp2an 654 . . . . . . . 8
15 eluniab 4027 . . . . . . . 8
1614, 15mpbir 201 . . . . . . 7
1716, 22th 231 . . . . . 6
1817eqriv 2433 . . . . 5
1918eleq1i 2499 . . . 4
201, 19mtbir 291 . . 3
21 uniexg 4706 . . 3
2220, 21mto 169 . 2
2322nelir 2698 1
 Colors of variables: wff set class Syntax hints:   wa 359  wex 1550   wceq 1652   wcel 1725  cab 2422   wnel 2600  cvv 2956  csn 3814  cuni 4015 This theorem is referenced by:  fiprc  7188 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-rex 2711  df-v 2958  df-dif 3323  df-un 3325  df-nul 3629  df-sn 3820  df-pr 3821  df-uni 4016
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