| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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.) (Revised by AV, 12-Jun-2022.) |
| Ref | Expression |
|---|---|
| preqsn.1 | ⊢ 𝐴 ∈ V |
| preqsn.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| preqsn | ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | preqsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 2 | id 22 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
| 3 | preqsn.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
| 4 | 3 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ V → 𝐵 ∈ V) |
| 5 | 2, 4 | preqsnd 4859 | . . 3 ⊢ (𝐴 ∈ V → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
| 6 | 1, 5 | ax-mp 5 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
| 7 | eqeq2 2749 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶)) | |
| 8 | 7 | pm5.32ri 575 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
| 9 | 6, 8 | bitr4i 278 | 1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: opeqsng 5508 propeqop 5512 propssopi 5513 relop 5861 hash2prde 14509 symg2bas 19410 |
| Copyright terms: Public domain | W3C validator |