ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exsnrex GIF version

Theorem exsnrex 3649
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} ↔ ∃𝑥𝑀 𝑀 = {𝑥})

Proof of Theorem exsnrex
StepHypRef Expression
1 vex 2755 . . . . . 6 𝑥 ∈ V
21snid 3638 . . . . 5 𝑥 ∈ {𝑥}
3 eleq2 2253 . . . . 5 (𝑀 = {𝑥} → (𝑥𝑀𝑥 ∈ {𝑥}))
42, 3mpbiri 168 . . . 4 (𝑀 = {𝑥} → 𝑥𝑀)
54pm4.71ri 392 . . 3 (𝑀 = {𝑥} ↔ (𝑥𝑀𝑀 = {𝑥}))
65exbii 1616 . 2 (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥(𝑥𝑀𝑀 = {𝑥}))
7 df-rex 2474 . 2 (∃𝑥𝑀 𝑀 = {𝑥} ↔ ∃𝑥(𝑥𝑀𝑀 = {𝑥}))
86, 7bitr4i 187 1 (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥𝑀 𝑀 = {𝑥})
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wrex 2469  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sn 3613
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator