Theorem bj-elsngl 36970

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 2816	. 2 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵))
2		df-bj-sngl 36968	. . . . 5 ⊢ sngl 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}}
3	2	eqabri 2884	. . . 4 ⊢ (𝑦 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥})
4	3	anbi2i 623	. . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
5	4	exbii 1847	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
6		r19.42v 3190	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ (𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
7	6	bicomi 224	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
8	7	exbii 1847	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}) ↔ ∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
9		rexcom4 3287	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ ∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
10	9	bicomi 224	. . 3 ⊢ (∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
11		eqcom 2743	. . . . . 6 ⊢ (𝐴 = {𝑥} ↔ {𝑥} = 𝐴)
12		vsnex 5433	. . . . . . 7 ⊢ {𝑥} ∈ V
13	12	eqvinc 3648	. . . . . 6 ⊢ ({𝑥} = 𝐴 ↔ ∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴))
14		exancom 1860	. . . . . 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 224	. . . 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 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃wrex 3069  {csn 4625  sngl bj-csngl 36967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rex 3070  df-v 3481  df-un 3955  df-sn 4626  df-pr 4628  df-bj-sngl 36968

This theorem is referenced by:  bj-snglc  36971  bj-snglss  36972  bj-0nelsngl  36973  bj-eltag  36979