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Theorem funsneqopsn 6457
Description: A singleton of an ordered pair is an ordered pair of equal singletons if the components are equal. (Contributed by AV, 24-Sep-2020.)
Hypotheses
Ref Expression
funsndifnop.a 𝐴 ∈ V
funsndifnop.b 𝐵 ∈ V
funsndifnop.g 𝐺 = {⟨𝐴, 𝐵⟩}
Assertion
Ref Expression
funsneqopsn (𝐴 = 𝐵𝐺 = ⟨{𝐴}, {𝐴}⟩)

Proof of Theorem funsneqopsn
StepHypRef Expression
1 opeq2 4434 . . . 4 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
21sneqd 4222 . . 3 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
3 funsndifnop.g . . 3 𝐺 = {⟨𝐴, 𝐵⟩}
42, 3syl6reqr 2704 . 2 (𝐴 = 𝐵𝐺 = {⟨𝐴, 𝐴⟩})
5 eqid 2651 . . . 4 𝐴 = 𝐴
6 eqid 2651 . . . 4 {𝐴} = {𝐴}
75, 6, 63pm3.2i 1259 . . 3 (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})
8 eqeq1 2655 . . . 4 (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩))
9 funsndifnop.a . . . . 5 𝐴 ∈ V
10 snex 4938 . . . . 5 {𝐴} ∈ V
119, 9, 10, 10snopeqop 4998 . . . 4 ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
128, 11syl6bb 276 . . 3 (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})))
137, 12mpbiri 248 . 2 (𝐺 = {⟨𝐴, 𝐴⟩} → 𝐺 = ⟨{𝐴}, {𝐴}⟩)
144, 13syl 17 1 (𝐴 = 𝐵𝐺 = ⟨{𝐴}, {𝐴}⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cop 4216
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-3an 1056  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-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217
This theorem is referenced by:  funsneqop  6458  vtxvalsnop  25978  iedgvalsnop  25979
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