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Theorem bj-snglc 36153
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 3069 . 2 (∃𝑥𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
2 bj-elsngl 36152 . 2 ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥𝐵 {𝐴} = {𝑥})
3 elisset 2813 . . . . 5 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
43pm4.71i 558 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
5 19.42v 1955 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
6 eleq1 2819 . . . . . . 7 (𝐴 = 𝑥 → (𝐴𝐵𝑥𝐵))
76eqcoms 2738 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝐵𝑥𝐵))
87pm5.32ri 574 . . . . 5 ((𝐴𝐵𝑥 = 𝐴) ↔ (𝑥𝐵𝑥 = 𝐴))
98exbii 1848 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
104, 5, 93bitr2i 298 . . 3 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
11 sneqbg 4843 . . . . . . 7 (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴))
1211elv 3478 . . . . . 6 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
13 eqcom 2737 . . . . . 6 ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥})
1412, 13bitr3i 276 . . . . 5 (𝑥 = 𝐴 ↔ {𝐴} = {𝑥})
1514anbi2i 621 . . . 4 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ {𝐴} = {𝑥}))
1615exbii 1848 . . 3 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
1710, 16bitri 274 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
181, 2, 173bitr4ri 303 1 (𝐴𝐵 ↔ {𝐴} ∈ sngl 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wrex 3068  Vcvv 3472  {csn 4627  sngl bj-csngl 36149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rex 3069  df-v 3474  df-un 3952  df-sn 4628  df-pr 4630  df-bj-sngl 36150
This theorem is referenced by:  bj-snglinv  36156  bj-tagci  36168  bj-tagcg  36169
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