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Mirrors > Home > MPE Home > Th. List > preqsn | Structured version Visualization version GIF version |
Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Proof shortened by JJ, 23-Jul-2021.) |
Ref | Expression |
---|---|
preqsn.1 | ⊢ 𝐴 ∈ V |
preqsn.2 | ⊢ 𝐵 ∈ V |
preqsn.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
preqsn | ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4138 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
2 | 1 | eqeq2i 2622 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶}) |
3 | oridm 535 | . . 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 | preqsn.3 | . . . 4 ⊢ 𝐶 ∈ V | |
7 | 4, 5, 6, 6 | preq12b 4322 | . . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
8 | eqeq2 2621 | . . . 4 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶)) | |
9 | 8 | pm5.32ri 668 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
10 | 3, 7, 9 | 3bitr4i 291 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
11 | 2, 10 | bitri 263 | 1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 {cpr 4127 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-v 3175 df-un 3545 df-sn 4126 df-pr 4128 |
This theorem is referenced by: opeqsn 4892 propeqop 4895 propssopi 4896 relop 5194 hash2prde 13109 symg2bas 17641 |
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