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Theorem bj-snglc 33082
Description: Characterization of the elements of 𝐴 in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-snglc (𝐴𝐵 ↔ {𝐴} ∈ sngl 𝐵)

Proof of Theorem bj-snglc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-rex 2947 . 2 (∃𝑥𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
2 bj-elsngl 33081 . 2 ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥𝐵 {𝐴} = {𝑥})
3 elisset 3246 . . . . 5 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
43pm4.71i 665 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
5 19.42v 1921 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
6 eleq1 2718 . . . . . . 7 (𝐴 = 𝑥 → (𝐴𝐵𝑥𝐵))
76eqcoms 2659 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝐵𝑥𝐵))
87pm5.32ri 671 . . . . 5 ((𝐴𝐵𝑥 = 𝐴) ↔ (𝑥𝐵𝑥 = 𝐴))
98exbii 1814 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
104, 5, 93bitr2i 288 . . 3 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
11 vex 3234 . . . . . . 7 𝑥 ∈ V
12 sneqbg 4406 . . . . . . 7 (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴))
1311, 12ax-mp 5 . . . . . 6 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
14 eqcom 2658 . . . . . 6 ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥})
1513, 14bitr3i 266 . . . . 5 (𝑥 = 𝐴 ↔ {𝐴} = {𝑥})
1615anbi2i 730 . . . 4 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ {𝐴} = {𝑥}))
1716exbii 1814 . . 3 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
1810, 17bitri 264 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
191, 2, 183bitr4ri 293 1 (𝐴𝐵 ↔ {𝐴} ∈ sngl 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  {csn 4210  sngl bj-csngl 33078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213  df-bj-sngl 33079
This theorem is referenced by:  bj-snglinv  33085  bj-tagci  33097  bj-tagcg  33098
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