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Theorem bj-elsngl 35849
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 2812 . 2 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ sngl 𝐵))
2 df-bj-sngl 35847 . . . . 5 sngl 𝐵 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = {𝑥}}
32eqabri 2878 . . . 4 (𝑦 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝑦 = {𝑥})
43anbi2i 624 . . 3 ((𝑦 = 𝐴𝑦 ∈ sngl 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
54exbii 1851 . 2 (∃𝑦(𝑦 = 𝐴𝑦 ∈ sngl 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
6 r19.42v 3191 . . . . 5 (∃𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}) ↔ (𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
76bicomi 223 . . . 4 ((𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
87exbii 1851 . . 3 (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
9 rexcom4 3286 . . . 4 (∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
109bicomi 223 . . 3 (∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
11 eqcom 2740 . . . . . 6 (𝐴 = {𝑥} ↔ {𝑥} = 𝐴)
12 vsnex 5430 . . . . . . 7 {𝑥} ∈ V
1312eqvinc 3638 . . . . . 6 ({𝑥} = 𝐴 ↔ ∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴))
14 exancom 1865 . . . . . 6 (∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴) ↔ ∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
1511, 13, 143bitri 297 . . . . 5 (𝐴 = {𝑥} ↔ ∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
1615bicomi 223 . . . 4 (∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ 𝐴 = {𝑥})
1716rexbii 3095 . . 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 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wrex 3071  {csn 4629  sngl bj-csngl 35846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632  df-bj-sngl 35847
This theorem is referenced by:  bj-snglc  35850  bj-snglss  35851  bj-0nelsngl  35852  bj-eltag  35858
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