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Mirrors > Home > ILE Home > Th. List > preqsn | GIF version |
Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
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 3389 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
2 | 1 | eqeq2i 2050 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶}) |
3 | preqsn.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | preqsn.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | preqsn.3 | . . . 4 ⊢ 𝐶 ∈ V | |
6 | 3, 4, 5, 5 | preq12b 3541 | . . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
7 | oridm 674 | . . . 4 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) | |
8 | eqtr3 2059 | . . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) | |
9 | simpr 103 | . . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶) | |
10 | 8, 9 | jca 290 | . . . . 5 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
11 | eqtr 2057 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶) | |
12 | simpr 103 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶) | |
13 | 11, 12 | jca 290 | . . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
14 | 10, 13 | impbii 117 | . . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
15 | 7, 14 | bitri 173 | . . 3 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
16 | 6, 15 | bitri 173 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
17 | 2, 16 | bitri 173 | 1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∨ wo 629 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {csn 3375 {cpr 3376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 |
This theorem is referenced by: opeqsn 3989 relop 4486 |
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