Theorem exsnrex 4168

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 4156	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
2		eleq2 2677	. . . . 5 ⊢ (𝑀 = {𝑥} → (𝑥 ∈ 𝑀 ↔ 𝑥 ∈ {𝑥}))
3	1, 2	mpbiri 247	. . . 4 ⊢ (𝑀 = {𝑥} → 𝑥 ∈ 𝑀)
4	3	pm4.71ri 663	. . 3 ⊢ (𝑀 = {𝑥} ↔ (𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
5	4	exbii 1764	. 2 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
6		df-rex 2902	. 2 ⊢ (∃𝑥 ∈ 𝑀 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
7	5, 6	bitr4i 266	1 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥})

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sn 4126

This theorem is referenced by:  frgrawopreg1  26577  frgrawopreg2  26578  frgrwopreg1  41487  frgrwopreg2  41488