Theorem bj-elsngl 37699

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 2838	. 2 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵))
2		df-bj-sngl 37697	. . . . 5 ⊢ sngl 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}}
3	2	eqabri 2904	. . . 4 ⊢ (𝑦 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥})
4	3	anbi2i 635	. . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
5	4	exbii 1881	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ sngl 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
6		r19.42v 3196	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ (𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}))
7	6	bicomi 227	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
8	7	exbii 1881	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}) ↔ ∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
9		rexcom4 3291	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ ∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
10	9	bicomi 227	. . 3 ⊢ (∃𝑦∃𝑥 ∈ 𝐵 (𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
11		eqcom 2769	. . . . . 6 ⊢ (𝐴 = {𝑥} ↔ {𝑥} = 𝐴)
12		vsnex 5404	. . . . . . 7 ⊢ {𝑥} ∈ V
13	12	eqvinc 3606	. . . . . 6 ⊢ ({𝑥} = 𝐴 ↔ ∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴))
14		exancom 1894	. . . . . 6 ⊢ (∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
15	11, 13, 14	3bitri 300	. . . . 5 ⊢ (𝐴 = {𝑥} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}))
16	15	bicomi 227	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ 𝐴 = {𝑥})
17	16	rexbii 3111	. . 3 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
18	8, 10, 17	3bitri 300	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥 ∈ 𝐵 𝑦 = {𝑥}) ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
19	1, 5, 18	3bitri 300	1 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088  {csn 4587  sngl bj-csngl 37696

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3455  df-un 3907  df-sn 4588  df-pr 4590  df-bj-sngl 37697

This theorem is used by:  bj-snglc  37700  bj-snglss  37701  bj-0nelsngl  37702  bj-eltag  37708