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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 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 4577 | . . 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 2810 | . . 3 ⊢ (𝐵 = 𝐶 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶)) | |
8 | 7 | pm5.32ri 572 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
9 | 6, 8 | bitr4i 270 | 1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3385 {csn 4368 {cpr 4370 |
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 |
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-v 3387 df-dif 3772 df-un 3774 df-nul 4116 df-sn 4369 df-pr 4371 |
This theorem is referenced by: opeqsng 5157 opeqsnOLD 5159 propeqop 5163 propssopi 5164 relop 5476 hash2prde 13501 symg2bas 18130 |
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