Theorem bj-snglc 37633

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 3089	. 2 ⊢ (∃𝑥 ∈ 𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥}))
2		bj-elsngl 37632	. 2 ⊢ ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 {𝐴} = {𝑥})
3		elisset 2844	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
4	3	pm4.71i 568	. . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
5		19.42v 1982	. . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
6		eleq1 2850	. . . . . . 7 ⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
7	6	eqcoms 2770	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
8	7	pm5.32ri 585	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
9	8	exbii 1877	. . . 4 ⊢ (∃𝑥(𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
10	4, 5, 9	3bitr2i 302	. . 3 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
11		sneqbg 4807	. . . . . . 7 ⊢ (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴))
12	11	elv 3459	. . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
13		eqcom 2769	. . . . . 6 ⊢ ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥})
14	12, 13	bitr3i 280	. . . . 5 ⊢ (𝑥 = 𝐴 ↔ {𝐴} = {𝑥})
15	14	anbi2i 634	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥}))
16	15	exbii 1877	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥}))
17	10, 16	bitri 278	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ {𝐴} = {𝑥}))
18	1, 2, 17	3bitr4ri 307	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ sngl 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ∃wrex 3088  Vcvv 3454  {csn 4588  sngl bj-csngl 37629

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3456  df-un 3909  df-sn 4589  df-pr 4591  df-bj-sngl 37630

This theorem is used by:  bj-snglinv  37636  bj-tagci  37648  bj-tagcg  37649