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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 ↔ ∃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 dfclel 2816 . 2 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ sngl 𝐵))
2 df-bj-sngl 36968 . . . . 5 sngl 𝐵 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = {𝑥}}
32eqabri 2884 . . . 4 (𝑦 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝑦 = {𝑥})
43anbi2i 623 . . 3 ((𝑦 = 𝐴𝑦 ∈ sngl 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
54exbii 1847 . 2 (∃𝑦(𝑦 = 𝐴𝑦 ∈ sngl 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
6 r19.42v 3190 . . . . 5 (∃𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}) ↔ (𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
76bicomi 224 . . . 4 ((𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
87exbii 1847 . . 3 (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
9 rexcom4 3287 . . . 4 (∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
109bicomi 224 . . 3 (∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
11 eqcom 2743 . . . . . 6 (𝐴 = {𝑥} ↔ {𝑥} = 𝐴)
12 vsnex 5433 . . . . . . 7 {𝑥} ∈ V
1312eqvinc 3648 . . . . . 6 ({𝑥} = 𝐴 ↔ ∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴))
14 exancom 1860 . . . . . 6 (∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴) ↔ ∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
1511, 13, 143bitri 297 . . . . 5 (𝐴 = {𝑥} ↔ ∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
1615bicomi 224 . . . 4 (∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ 𝐴 = {𝑥})
1716rexbii 3093 . . 3 (∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑥𝐵 𝐴 = {𝑥})
188, 10, 173bitri 297 . 2 (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑥𝐵 𝐴 = {𝑥})
191, 5, 183bitri 297 1 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
Colors of variables: wff setvar class
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
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