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Theorem bj-snglc 36951
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 3068 . 2 (∃𝑥𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
2 bj-elsngl 36950 . 2 ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥𝐵 {𝐴} = {𝑥})
3 elisset 2820 . . . . 5 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
43pm4.71i 559 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
5 19.42v 1950 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
6 eleq1 2826 . . . . . . 7 (𝐴 = 𝑥 → (𝐴𝐵𝑥𝐵))
76eqcoms 2742 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝐵𝑥𝐵))
87pm5.32ri 575 . . . . 5 ((𝐴𝐵𝑥 = 𝐴) ↔ (𝑥𝐵𝑥 = 𝐴))
98exbii 1844 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
104, 5, 93bitr2i 299 . . 3 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
11 sneqbg 4847 . . . . . . 7 (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴))
1211elv 3482 . . . . . 6 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
13 eqcom 2741 . . . . . 6 ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥})
1412, 13bitr3i 277 . . . . 5 (𝑥 = 𝐴 ↔ {𝐴} = {𝑥})
1514anbi2i 623 . . . 4 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ {𝐴} = {𝑥}))
1615exbii 1844 . . 3 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
1710, 16bitri 275 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
181, 2, 173bitr4ri 304 1 (𝐴𝐵 ↔ {𝐴} ∈ sngl 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wrex 3067  Vcvv 3477  {csn 4630  sngl bj-csngl 36947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rex 3068  df-v 3479  df-un 3967  df-sn 4631  df-pr 4633  df-bj-sngl 36948
This theorem is referenced by:  bj-snglinv  36954  bj-tagci  36966  bj-tagcg  36967
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