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Mirrors > Home > MPE Home > Th. List > elopg | Structured version Visualization version GIF version |
Description: Characterization of the elements of an ordered pair. Closed form of elop 5478. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
elopg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 〈𝐴, 𝐵〉 ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 4876 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
2 | eleq2 2828 | . . 3 ⊢ (〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ 〈𝐴, 𝐵〉 ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}})) | |
3 | snex 5442 | . . . 4 ⊢ {𝐴} ∈ V | |
4 | prex 5443 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
5 | 3, 4 | elpr2 4657 | . . 3 ⊢ (𝐶 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})) |
6 | 2, 5 | bitrdi 287 | . 2 ⊢ (〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ 〈𝐴, 𝐵〉 ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))) |
7 | 1, 6 | syl 17 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 〈𝐴, 𝐵〉 ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 {csn 4631 {cpr 4633 〈cop 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: elop 5478 bj-inftyexpidisj 37193 |
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