Theorem funsneqopsnOLD 6645

Description: Obsolete as of 15-Jul-2022. Use snopeqopsnid 5165 instead. (Contributed by AV, 24-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
funsndifnop.a	⊢ 𝐴 ∈ V
funsndifnop.b	⊢ 𝐵 ∈ V
funsndifnop.g	⊢ 𝐺 = {⟨𝐴, 𝐵⟩}

Assertion

Ref	Expression
funsneqopsnOLD	⊢ (𝐴 = 𝐵 → 𝐺 = ⟨{𝐴}, {𝐴}⟩)

Proof of Theorem funsneqopsnOLD

Step	Hyp	Ref	Expression
1		opeq2 4594	. . . 4 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
2	1	sneqd 4380	. . 3 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
3		funsndifnop.g	. . 3 ⊢ 𝐺 = {⟨𝐴, 𝐵⟩}
4	2, 3	syl6reqr 2852	. 2 ⊢ (𝐴 = 𝐵 → 𝐺 = {⟨𝐴, 𝐴⟩})
5		eqid 2799	. . . 4 ⊢ 𝐴 = 𝐴
6		eqid 2799	. . . 4 ⊢ {𝐴} = {𝐴}
7	5, 6, 6	3pm3.2i 1439	. . 3 ⊢ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})
8		eqeq1 2803	. . . 4 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩))
9		funsndifnop.a	. . . . 5 ⊢ 𝐴 ∈ V
10		snex 5099	. . . . 5 ⊢ {𝐴} ∈ V
11	9, 9, 10, 10	snopeqopOLD 5162	. . . 4 ⊢ ({⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴}))
12	8, 11	syl6bb 279	. . 3 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → (𝐺 = ⟨{𝐴}, {𝐴}⟩ ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})))
13	7, 12	mpbiri 250	. 2 ⊢ (𝐺 = {⟨𝐴, 𝐴⟩} → 𝐺 = ⟨{𝐴}, {𝐴}⟩)
14	4, 13	syl 17	1 ⊢ (𝐴 = 𝐵 → 𝐺 = ⟨{𝐴}, {𝐴}⟩)

Syntax hints:   → wi 4   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  Vcvv 3385  {csn 4368  ⟨cop 4374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375

This theorem is referenced by:  funsneqopOLD  6646