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| Description: Two classes related by an ordered-pair class abstraction are sets. (Contributed by AV, 21-Jan-2020.) | 
| Ref | Expression | 
|---|---|
| bropaex12.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓} | 
| Ref | Expression | 
|---|---|
| bropaex12 | ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-br 5144 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 2 | bropaex12.1 | . . . . 5 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓} | |
| 3 | 2 | eleq2i 2833 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) | 
| 4 | 1, 3 | bitri 275 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) | 
| 5 | elopaelxp 5775 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
| 6 | 4, 5 | sylbi 217 | . 2 ⊢ (𝐴𝑅𝐵 → 〈𝐴, 𝐵〉 ∈ (V × V)) | 
| 7 | opelxp 5721 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
| 8 | 6, 7 | sylib 218 | 1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 class class class wbr 5143 {copab 5205 × cxp 5683 | 
| 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-ral 3062 df-rex 3071 df-rab 3437 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 df-br 5144 df-opab 5206 df-xp 5691 | 
| This theorem is referenced by: fpwwe 10686 efgrelexlema 19767 brsslt 27830 rgrprop 29578 rusgrprop 29580 bropabg 43336 clcllaw 48107 asslawass 48109 | 
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