![]() |
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 3713 | . . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐶} |
3 | eleq2 2253 | . . . 4 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶})) | |
4 | 2, 3 | mpbii 148 | . . 3 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}) |
5 | 1 | elpr 3628 | . . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
6 | 4, 5 | sylib 122 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
7 | preqr1.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
8 | 7 | prid1 3713 | . . . 4 ⊢ 𝐵 ∈ {𝐵, 𝐶} |
9 | eleq2 2253 | . . . 4 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶})) | |
10 | 8, 9 | mpbiri 168 | . . 3 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}) |
11 | 7 | elpr 3628 | . . 3 ⊢ (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) |
12 | 10, 11 | sylib 122 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)) |
13 | eqcom 2191 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
14 | eqeq2 2199 | . 2 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶)) | |
15 | 6, 12, 13, 14 | oplem1 977 | 1 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 709 = wceq 1364 ∈ wcel 2160 Vcvv 2752 {cpr 3608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 |
This theorem is referenced by: preqr2 3784 |
Copyright terms: Public domain | W3C validator |