Theorem bj-elsngl 36165

Description: Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-elsngl	⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-elsngl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfclel 2810	. 2 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵))
2		df-bj-sngl 36163	. . . . 5 ⊢ sngl 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}}
3	2	eqabri 2876	. . . 4 ⊢ (𝑦 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥})
4	3	anbi2i 622	. . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
5	4	exbii 1849	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
6		r19.42v 3189	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ (𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
7	6	bicomi 223	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
8	7	exbii 1849	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}) ↔ ∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
9		rexcom4 3284	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ ∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
10	9	bicomi 223	. . 3 ⊢ (∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
11		eqcom 2738	. . . . . 6 ⊢ (𝐴 = {𝑥} ↔ {𝑥} = 𝐴)
12		vsnex 5429	. . . . . . 7 ⊢ {𝑥} ∈ V
13	12	eqvinc 3637	. . . . . 6 ⊢ ({𝑥} = 𝐴 ↔ ∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴))
14		exancom 1863	. . . . . 6 ⊢ (∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
15	11, 13, 14	3bitri 297	. . . . 5 ⊢ (𝐴 = {𝑥} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
16	15	bicomi 223	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ 𝐴 = {𝑥})
17	16	rexbii 3093	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
18	8, 10, 17	3bitri 297	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
19	1, 5, 18	3bitri 297	1 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∃wrex 3069  {csn 4628  sngl bj-csngl 36162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3475  df-un 3953  df-sn 4629  df-pr 4631  df-bj-sngl 36163

This theorem is referenced by:  bj-snglc  36166  bj-snglss  36167  bj-0nelsngl  36168  bj-eltag  36174