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 3548 | . . 3 ⊢ 〈𝐵, 𝐶〉 = {{𝐵}, {𝐵, 𝐶}} |
4 | 3 | eleq2i 2104 | . 2 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}}) |
5 | elop.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | elpr 3396 | . 2 ⊢ (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
7 | 4, 6 | bitri 173 | 1 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∨ wo 629 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {csn 3375 {cpr 3376 〈cop 3378 |
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-3an 887 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 df-op 3384 |
This theorem is referenced by: relop 4486 bdop 9995 |
Copyright terms: Public domain | W3C validator |