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Theorem bj-elsngl 32149
 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 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-elsngl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-clel 2606 . 2 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ sngl 𝐵))
2 df-bj-sngl 32147 . . . . 5 sngl 𝐵 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = {𝑥}}
32abeq2i 2722 . . . 4 (𝑦 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝑦 = {𝑥})
43anbi2i 726 . . 3 ((𝑦 = 𝐴𝑦 ∈ sngl 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
54exbii 1764 . 2 (∃𝑦(𝑦 = 𝐴𝑦 ∈ sngl 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
6 r19.42v 3073 . . . . 5 (∃𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}) ↔ (𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
76bicomi 213 . . . 4 ((𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
87exbii 1764 . . 3 (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
9 rexcom4 3198 . . . 4 (∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
109bicomi 213 . . 3 (∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
11 eqcom 2617 . . . . . 6 (𝐴 = {𝑥} ↔ {𝑥} = 𝐴)
12 snex 4835 . . . . . . 7 {𝑥} ∈ V
1312eqvinc 3300 . . . . . 6 ({𝑥} = 𝐴 ↔ ∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴))
14 exancom 1774 . . . . . 6 (∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴) ↔ ∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
1511, 13, 143bitri 285 . . . . 5 (𝐴 = {𝑥} ↔ ∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
1615bicomi 213 . . . 4 (∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ 𝐴 = {𝑥})
1716rexbii 3023 . . 3 (∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑥𝐵 𝐴 = {𝑥})
188, 10, 173bitri 285 . 2 (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑥𝐵 𝐴 = {𝑥})
191, 5, 183bitri 285 1 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {csn 4125  sngl bj-csngl 32146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-bj-sngl 32147 This theorem is referenced by:  bj-snglc  32150  bj-snglss  32151  bj-0nelsngl  32152  bj-eltag  32158
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