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 = 𝐵 → 𝐺 = ⟨{𝐴}, {𝐴}⟩)

Proof of Theorem funsneqopsn

Step	Hyp	Ref	Expression
1		opeq2 4434	. . . 4 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
2	1	sneqd 4222	. . 3 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
3		funsndifnop.g	. . 3 ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}
4	2, 3	syl6reqr 2704	. 2 ⊢ (𝐴 = 𝐵 → 𝐺 = {⟨𝐴, 𝐴⟩})
5		eqid 2651	. . . 4 ⊢ 𝐴 = 𝐴
6		eqid 2651	. . . 4 ⊢ {𝐴} = {𝐴}
7	5, 6, 6	3pm3.2i 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
11	9, 9, 10, 10	snopeqop 4998	. . . 4 ⊢ ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
12	8, 11	syl6bb 276	. . 3 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})))
13	7, 12	mpbiri 248	. 2 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → 𝐺 = ⟨{𝐴}, {𝐴}⟩)
14	4, 13	syl 17	1 ⊢ (𝐴 = 𝐵 → 𝐺 = ⟨{𝐴}, {𝐴}⟩)

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