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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.) (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 4783 | . . 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 2833 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶)) | |
8 | 7 | pm5.32ri 576 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
9 | 6, 8 | bitr4i 279 | 1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3495 {csn 4559 {cpr 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3497 df-dif 3938 df-un 3940 df-nul 4291 df-sn 4560 df-pr 4562 |
This theorem is referenced by: opeqsng 5385 propeqop 5389 propssopi 5390 relop 5715 hash2prde 13818 symg2bas 18457 |
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