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Theorem bj-elsngl 34268
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 2892 . 2 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ sngl 𝐵))
2 df-bj-sngl 34266 . . . . 5 sngl 𝐵 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = {𝑥}}
32abeq2i 2946 . . . 4 (𝑦 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝑦 = {𝑥})
43anbi2i 624 . . 3 ((𝑦 = 𝐴𝑦 ∈ sngl 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
54exbii 1841 . 2 (∃𝑦(𝑦 = 𝐴𝑦 ∈ sngl 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
6 r19.42v 3348 . . . . 5 (∃𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}) ↔ (𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}))
76bicomi 226 . . . 4 ((𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
87exbii 1841 . . 3 (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
9 rexcom4 3247 . . . 4 (∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}))
109bicomi 226 . . 3 (∃𝑦𝑥𝐵 (𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
11 eqcom 2826 . . . . . 6 (𝐴 = {𝑥} ↔ {𝑥} = 𝐴)
12 snex 5322 . . . . . . 7 {𝑥} ∈ V
1312eqvinc 3640 . . . . . 6 ({𝑥} = 𝐴 ↔ ∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴))
14 exancom 1854 . . . . . 6 (∃𝑦(𝑦 = {𝑥} ∧ 𝑦 = 𝐴) ↔ ∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
1511, 13, 143bitri 299 . . . . 5 (𝐴 = {𝑥} ↔ ∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}))
1615bicomi 226 . . . 4 (∃𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ 𝐴 = {𝑥})
1716rexbii 3245 . . 3 (∃𝑥𝐵𝑦(𝑦 = 𝐴𝑦 = {𝑥}) ↔ ∃𝑥𝐵 𝐴 = {𝑥})
188, 10, 173bitri 299 . 2 (∃𝑦(𝑦 = 𝐴 ∧ ∃𝑥𝐵 𝑦 = {𝑥}) ↔ ∃𝑥𝐵 𝐴 = {𝑥})
191, 5, 183bitri 299 1 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1530  wex 1773  wcel 2107  wrex 3137  {csn 4559  sngl bj-csngl 34265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rex 3142  df-v 3495  df-dif 3937  df-un 3939  df-nul 4290  df-sn 4560  df-pr 4562  df-bj-sngl 34266
This theorem is referenced by:  bj-snglc  34269  bj-snglss  34270  bj-0nelsngl  34271  bj-eltag  34277
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