Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3699 | . . 3 ⊢ 〈𝐵, 𝐶〉 = {{𝐵}, {𝐵, 𝐶}} |
4 | 3 | eleq2i 2204 | . 2 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}}) |
5 | elop.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | elpr 3543 | . 2 ⊢ (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
7 | 4, 6 | bitri 183 | 1 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 Vcvv 2681 {csn 3522 {cpr 3523 〈cop 3525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 |
This theorem is referenced by: relop 4684 bdop 13062 |
Copyright terms: Public domain | W3C validator |