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Theorem exsnrex 3383
Description: There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
Assertion
Ref Expression
exsnrex (x 𝑀 = {x} ↔ x 𝑀 𝑀 = {x})

Proof of Theorem exsnrex
StepHypRef Expression
1 vex 2534 . . . . . 6 x V
21snid 3373 . . . . 5 x {x}
3 eleq2 2079 . . . . 5 (𝑀 = {x} → (x 𝑀x {x}))
42, 3mpbiri 157 . . . 4 (𝑀 = {x} → x 𝑀)
54pm4.71ri 372 . . 3 (𝑀 = {x} ↔ (x 𝑀 𝑀 = {x}))
65exbii 1474 . 2 (x 𝑀 = {x} ↔ x(x 𝑀 𝑀 = {x}))
7 df-rex 2286 . 2 (x 𝑀 𝑀 = {x} ↔ x(x 𝑀 𝑀 = {x}))
86, 7bitr4i 176 1 (x 𝑀 = {x} ↔ x 𝑀 𝑀 = {x})
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1226  wex 1358   wcel 1370  wrex 2281  {csn 3346
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sn 3352
This theorem is referenced by: (None)
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