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Theorem exsnrex 4610
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 vsnid 4594 . . . . 5 𝑥 ∈ {𝑥}
2 eleq2 2899 . . . . 5 (𝑀 = {𝑥} → (𝑥𝑀𝑥 ∈ {𝑥}))
31, 2mpbiri 260 . . . 4 (𝑀 = {𝑥} → 𝑥𝑀)
43pm4.71ri 563 . . 3 (𝑀 = {𝑥} ↔ (𝑥𝑀𝑀 = {𝑥}))
54exbii 1842 . 2 (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥(𝑥𝑀𝑀 = {𝑥}))
6 df-rex 3142 . 2 (∃𝑥𝑀 𝑀 = {𝑥} ↔ ∃𝑥(𝑥𝑀𝑀 = {𝑥}))
75, 6bitr4i 280 1 (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥𝑀 𝑀 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1531  wex 1774  wcel 2108  wrex 3137  {csn 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rex 3142  df-v 3495  df-sn 4560
This theorem is referenced by:  frgrwopreg1  28089  frgrwopreg2  28090
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