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Mirrors > Home > MPE Home > Th. List > elop | Structured version Visualization version GIF version |
Description: Characterization of the elements of an ordered pair. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) Remove an extraneous hypothesis. (Revised by BJ, 25-Dec-2020.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
elop.1 | ⊢ 𝐵 ∈ V |
elop.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elop | ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elop.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | elop.2 | . 2 ⊢ 𝐶 ∈ V | |
3 | elopg 5422 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 Vcvv 3444 {csn 4585 {cpr 4587 〈cop 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 |
This theorem is referenced by: relop 5803 |
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