Theorem funsneqopsn 6322

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 4341	. . . 4 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
2	1	sneqd 4137	. . 3 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
3		funsndifnop.g	. . 3 ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}
4	2, 3	syl6reqr 2663	. 2 ⊢ (𝐴 = 𝐵 → 𝐺 = {⟨𝐴, 𝐴⟩})
5		eqid 2610	. . . 4 ⊢ 𝐴 = 𝐴
6		eqid 2610	. . . 4 ⊢ {𝐴} = {𝐴}
7	5, 6, 6	3pm3.2i 1232	. . 3 ⊢ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})
8		eqeq1 2614	. . . 4 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩))
9		funsndifnop.a	. . . . 5 ⊢ 𝐴 ∈ V
10		snex 4835	. . . . 5 ⊢ {𝐴} ∈ V
11	9, 9, 10, 10	snopeqop 4894	. . . 4 ⊢ ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
12	8, 11	syl6bb 275	. . 3 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})))
13	7, 12	mpbiri 247	. 2 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → 𝐺 = ⟨{𝐴}, {𝐴}⟩)
14	4, 13	syl 17	1 ⊢ (𝐴 = 𝐵 → 𝐺 = ⟨{𝐴}, {𝐴}⟩)

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132

This theorem is referenced by:  funsneqop  6323  vtxvalsnop  25716  iedgvalsnop  25717