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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 4816 | . . 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 2748 | . . 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 277 | 1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 {csn 4586 {cpr 4588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-v 3447 df-dif 3913 df-un 3915 df-nul 4283 df-sn 4587 df-pr 4589 |
This theorem is referenced by: opeqsng 5460 propeqop 5464 propssopi 5465 relop 5806 hash2prde 14369 symg2bas 19174 |
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