Theorem exsnrex 3566

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		vex 2689	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snid 3556	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
3		eleq2 2203	. . . . 5 ⊢ (𝑀 = {𝑥} → (𝑥 ∈ 𝑀 ↔ 𝑥 ∈ {𝑥}))
4	2, 3	mpbiri 167	. . . 4 ⊢ (𝑀 = {𝑥} → 𝑥 ∈ 𝑀)
5	4	pm4.71ri 389	. . 3 ⊢ (𝑀 = {𝑥} ↔ (𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
6	5	exbii 1584	. 2 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
7		df-rex 2422	. 2 ⊢ (∃𝑥 ∈ 𝑀 𝑀 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑀 ∧ 𝑀 = {𝑥}))
8	6, 7	bitr4i 186	1 ⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥})

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  {csn 3527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sn 3533

This theorem is referenced by: (None)