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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 3792 | . . 3 ⊢ 〈𝐵, 𝐶〉 = {{𝐵}, {𝐵, 𝐶}} |
4 | 3 | eleq2i 2256 | . 2 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}}) |
5 | elop.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | elpr 3628 | . 2 ⊢ (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
7 | 4, 6 | bitri 184 | 1 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∨ wo 709 = wceq 1364 ∈ wcel 2160 Vcvv 2752 {csn 3607 {cpr 3608 〈cop 3610 |
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-3an 982 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 df-op 3616 |
This theorem is referenced by: relop 4792 bdop 15064 |
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