Theorem funsneqopsn 6417

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 4403	. . . 4 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
2	1	sneqd 4189	. . 3 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
3		funsndifnop.g	. . 3 ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}
4	2, 3	syl6reqr 2675	. 2 ⊢ (𝐴 = 𝐵 → 𝐺 = {⟨𝐴, 𝐴⟩})
5		eqid 2622	. . . 4 ⊢ 𝐴 = 𝐴
6		eqid 2622	. . . 4 ⊢ {𝐴} = {𝐴}
7	5, 6, 6	3pm3.2i 1239	. . 3 ⊢ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})
8		eqeq1 2626	. . . 4 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩))
9		funsndifnop.a	. . . . 5 ⊢ 𝐴 ∈ V
10		snex 4908	. . . . 5 ⊢ {𝐴} ∈ V
11	9, 9, 10, 10	snopeqop 4969	. . . 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 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  funsneqop  6418  vtxvalsnop  25933  iedgvalsnop  25934