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Theorem bj-snglc 34276
 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 3144 . 2 (∃𝑥𝐵 {𝐴} = {𝑥} ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
2 bj-elsngl 34275 . 2 ({𝐴} ∈ sngl 𝐵 ↔ ∃𝑥𝐵 {𝐴} = {𝑥})
3 elisset 3505 . . . . 5 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
43pm4.71i 562 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
5 19.42v 1950 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ (𝐴𝐵 ∧ ∃𝑥 𝑥 = 𝐴))
6 eleq1 2900 . . . . . . 7 (𝐴 = 𝑥 → (𝐴𝐵𝑥𝐵))
76eqcoms 2829 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝐵𝑥𝐵))
87pm5.32ri 578 . . . . 5 ((𝐴𝐵𝑥 = 𝐴) ↔ (𝑥𝐵𝑥 = 𝐴))
98exbii 1844 . . . 4 (∃𝑥(𝐴𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
104, 5, 93bitr2i 301 . . 3 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝑥 = 𝐴))
11 sneqbg 4767 . . . . . . 7 (𝑥 ∈ V → ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴))
1211elv 3499 . . . . . 6 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
13 eqcom 2828 . . . . . 6 ({𝑥} = {𝐴} ↔ {𝐴} = {𝑥})
1412, 13bitr3i 279 . . . . 5 (𝑥 = 𝐴 ↔ {𝐴} = {𝑥})
1514anbi2i 624 . . . 4 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝑥𝐵 ∧ {𝐴} = {𝑥}))
1615exbii 1844 . . 3 (∃𝑥(𝑥𝐵𝑥 = 𝐴) ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
1710, 16bitri 277 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵 ∧ {𝐴} = {𝑥}))
181, 2, 173bitr4ri 306 1 (𝐴𝐵 ↔ {𝐴} ∈ sngl 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃wrex 3139  Vcvv 3494  {csn 4560  sngl bj-csngl 34272 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4561  df-pr 4563  df-bj-sngl 34273 This theorem is referenced by:  bj-snglinv  34279  bj-tagci  34291  bj-tagcg  34292
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