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Mirrors > Home > ILE Home > Th. List > preqr1 | GIF version |
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
preqr1.1 | ⊢ 𝐴 ∈ V |
preqr1.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
preqr1 | ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preqr1.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | 1 | prid1 3476 | . . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐶} |
3 | eleq2 2101 | . . . 4 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶})) | |
4 | 2, 3 | mpbii 136 | . . 3 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}) |
5 | 1 | elpr 3396 | . . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
6 | 4, 5 | sylib 127 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
7 | preqr1.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
8 | 7 | prid1 3476 | . . . 4 ⊢ 𝐵 ∈ {𝐵, 𝐶} |
9 | eleq2 2101 | . . . 4 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶})) | |
10 | 8, 9 | mpbiri 157 | . . 3 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}) |
11 | 7 | elpr 3396 | . . 3 ⊢ (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) |
12 | 10, 11 | sylib 127 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) |
13 | eqcom 2042 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
14 | eqeq2 2049 | . 2 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶)) | |
15 | 6, 12, 13, 14 | oplem1 882 | 1 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 629 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {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: preqr2 3540 |
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