Theorem exsnrex 4688

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

Step	Hyp	Ref	Expression
1		vsnid 4671	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
2		eleq2 2830	. . . . 5 ⊢ (𝑀 = {𝑥} → (𝑥 ∈ 𝑀 ↔ 𝑥 ∈ {𝑥}))
3	1, 2	mpbiri 258	. . . 4 ⊢ (𝑀 = {𝑥} → 𝑥 ∈ 𝑀)
4	3	pm4.71ri 560	. . 3 ⊢ (𝑀 = {𝑥} ↔ (𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
5	4	exbii 1847	. 2 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
6		df-rex 3071	. 2 ⊢ (∃𝑥 ∈ 𝑀 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
7	5, 6	bitr4i 278	1 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥})

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2108  ∃wrex 3070  {csn 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3483  df-sn 4635

This theorem is referenced by:  frgrwopreg1  30363  frgrwopreg2  30364