Theorem preqsnOLD 4529

Description: Obsolete proof of preqsn 4528 as of 12-Jun-2022. (Contributed by NM, 3-Jun-2008.) (Proof shortened by JJ, 23-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
preqsn.1	⊢ 𝐴 ∈ V
preqsn.2	⊢ 𝐵 ∈ V
preqsnOLD.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
preqsnOLD	⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

Proof of Theorem preqsnOLD

Step	Hyp	Ref	Expression
1		dfsn2 4330	. . 3 ⊢ {𝐶} = {𝐶, 𝐶}
2	1	eqeq2i 2783	. 2 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
3		oridm 882	. . 3 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
4		preqsn.1	. . . 4 ⊢ 𝐴 ∈ V
5		preqsn.2	. . . 4 ⊢ 𝐵 ∈ V
6		preqsnOLD.3	. . . 4 ⊢ 𝐶 ∈ V
7	4, 5, 6, 6	preq12b 4514	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
8		eqeq2 2782	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶))
9	8	pm5.32ri 559	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
10	3, 7, 9	3bitr4i 292	. 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
11	2, 10	bitri 264	1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

Syntax hints:   ↔ wb 196   ∧ wa 382   ∨ wo 828   = wceq 1631   ∈ wcel 2145  Vcvv 3351  {csn 4317  {cpr 4319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3729  df-sn 4318  df-pr 4320

This theorem is referenced by: (None)