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Theorem bj-snglc 33405
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 3061 . 2 (∃𝑥𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
2 bj-elsngl 33404 . 2 ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥𝐵 {𝐴} = {𝑥})
3 elisset 3368 . . . . 5 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
43pm4.71i 555 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
5 19.42v 2048 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
6 eleq1 2832 . . . . . . 7 (𝐴 = 𝑥 → (𝐴𝐵𝑥𝐵))
76eqcoms 2773 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝐵𝑥𝐵))
87pm5.32ri 571 . . . . 5 ((𝐴𝐵𝑥 = 𝐴) ↔ (𝑥𝐵𝑥 = 𝐴))
98exbii 1943 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
104, 5, 93bitr2i 290 . . 3 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
11 vex 3353 . . . . . . 7 𝑥 ∈ V
12 sneqbg 4528 . . . . . . 7 (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴))
1311, 12ax-mp 5 . . . . . 6 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
14 eqcom 2772 . . . . . 6 ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥})
1513, 14bitr3i 268 . . . . 5 (𝑥 = 𝐴 ↔ {𝐴} = {𝑥})
1615anbi2i 616 . . . 4 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ {𝐴} = {𝑥}))
1716exbii 1943 . . 3 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
1810, 17bitri 266 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
191, 2, 183bitr4ri 295 1 (𝐴𝐵 ↔ {𝐴} ∈ sngl 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wrex 3056  Vcvv 3350  {csn 4336  sngl bj-csngl 33401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3737  df-un 3739  df-nul 4082  df-sn 4337  df-pr 4339  df-bj-sngl 33402
This theorem is referenced by:  bj-snglinv  33408  bj-tagci  33420  bj-tagcg  33421
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