![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3568 | . . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐶} |
3 | eleq2 2158 | . . . 4 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶})) | |
4 | 2, 3 | mpbii 147 | . . 3 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}) |
5 | 1 | elpr 3487 | . . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
6 | 4, 5 | sylib 121 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
7 | preqr1.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
8 | 7 | prid1 3568 | . . . 4 ⊢ 𝐵 ∈ {𝐵, 𝐶} |
9 | eleq2 2158 | . . . 4 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶})) | |
10 | 8, 9 | mpbiri 167 | . . 3 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}) |
11 | 7 | elpr 3487 | . . 3 ⊢ (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) |
12 | 10, 11 | sylib 121 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) |
13 | eqcom 2097 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
14 | eqeq2 2104 | . 2 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶)) | |
15 | 6, 12, 13, 14 | oplem1 924 | 1 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 667 = wceq 1296 ∈ wcel 1445 Vcvv 2633 {cpr 3467 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-un 3017 df-sn 3472 df-pr 3473 |
This theorem is referenced by: preqr2 3635 |
Copyright terms: Public domain | W3C validator |