Theorem bj-snglc 35159

Description: Characterization of the elements of 𝐴 in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-snglc	⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵)

Proof of Theorem bj-snglc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 3070	. 2 ⊢ (∃𝑥 ∈ 𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥}))
2		bj-elsngl 35158	. 2 ⊢ ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 {𝐴} = {𝑥})
3		elisset 2820	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
4	3	pm4.71i 560	. . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
5		19.42v 1957	. . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
6		eleq1 2826	. . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
7	6	eqcoms 2746	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
8	7	pm5.32ri 576	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
9	8	exbii 1850	. . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
10	4, 5, 9	3bitr2i 299	. . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
11		sneqbg 4774	. . . . . . 7 ⊢ (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴))
12	11	elv 3438	. . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
13		eqcom 2745	. . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥})
14	12, 13	bitr3i 276	. . . . 5 ⊢ (𝑥 = 𝐴 ↔ {𝐴} = {𝑥})
15	14	anbi2i 623	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥}))
16	15	exbii 1850	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥}))
17	10, 16	bitri 274	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥}))
18	1, 2, 17	3bitr4ri 304	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432  {csn 4561  sngl bj-csngl 35155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564  df-bj-sngl 35156

This theorem is referenced by:  bj-snglinv  35162  bj-tagci  35174  bj-tagcg  35175