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| Description: Characterization of the elements of an ordered pair. Closed form of elop 5472. (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 4871 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
| 2 | eleq2 2830 | . . 3 ⊢ (〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ 〈𝐴, 𝐵〉 ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}})) | |
| 3 | snex 5436 | . . . 4 ⊢ {𝐴} ∈ V | |
| 4 | prex 5437 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
| 5 | 3, 4 | elpr2 4652 | . . 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 848 = wceq 1540 ∈ wcel 2108 {csn 4626 {cpr 4628 〈cop 4632 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 | 
| This theorem is referenced by: elop 5472 bj-inftyexpidisj 37211 | 
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