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Mirrors > Home > ILE Home > Th. List > elop | GIF version |
Description: An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
elop.1 | ⊢ 𝐴 ∈ V |
elop.2 | ⊢ 𝐵 ∈ V |
elop.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elop | ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elop.2 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | elop.3 | . . . 4 ⊢ 𝐶 ∈ V | |
3 | 1, 2 | dfop 3619 | . . 3 ⊢ 〈𝐵, 𝐶〉 = {{𝐵}, {𝐵, 𝐶}} |
4 | 3 | eleq2i 2154 | . 2 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}}) |
5 | elop.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | elpr 3465 | . 2 ⊢ (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
7 | 4, 6 | bitri 182 | 1 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∨ wo 664 = wceq 1289 ∈ wcel 1438 Vcvv 2619 {csn 3444 {cpr 3445 〈cop 3447 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-sn 3450 df-pr 3451 df-op 3453 |
This theorem is referenced by: relop 4582 bdop 11649 |
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