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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 4935. (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 4400 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
2 | eleq2 2690 | . . 3 ⊢ (〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ 〈𝐴, 𝐵〉 ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}})) | |
3 | snex 4908 | . . . 4 ⊢ {𝐴} ∈ V | |
4 | prex 4909 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
5 | 3, 4 | elpr2 4199 | . . 3 ⊢ (𝐶 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})) |
6 | 2, 5 | syl6bb 276 | . 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 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 {cpr 4179 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: elop 4935 bj-inftyexpidisj 33097 |
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