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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 5350. (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 4793 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
2 | eleq2 2899 | . . 3 ⊢ (〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ 〈𝐴, 𝐵〉 ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}})) | |
3 | snex 5322 | . . . 4 ⊢ {𝐴} ∈ V | |
4 | prex 5323 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
5 | 3, 4 | elpr2 4583 | . . 3 ⊢ (𝐶 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})) |
6 | 2, 5 | syl6bb 289 | . 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 208 ∧ wa 398 ∨ wo 843 = wceq 1531 ∈ wcel 2108 {csn 4559 {cpr 4561 〈cop 4565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 |
This theorem is referenced by: elop 5350 bj-inftyexpidisj 34484 |
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