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Theorem exsnrex 3404
 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 2554 . . . . . 6 x V
21snid 3394 . . . . 5 x {x}
3 eleq2 2098 . . . . 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 1493 . 2 (x 𝑀 = {x} ↔ x(x 𝑀 𝑀 = {x}))
7 df-rex 2306 . 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 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  {csn 3367 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sn 3373 This theorem is referenced by: (None)
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