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Mirrors > Home > MPE Home > Th. List > bropaex12 | Structured version Visualization version GIF version |
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 5149 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | bropaex12.1 | . . . . 5 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓} | |
3 | 2 | eleq2i 2831 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
4 | 1, 3 | bitri 275 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
5 | elopaelxp 5778 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
6 | 4, 5 | sylbi 217 | . 2 ⊢ (𝐴𝑅𝐵 → 〈𝐴, 𝐵〉 ∈ (V × V)) |
7 | opelxp 5725 | . 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 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 class class class wbr 5148 {copab 5210 × cxp 5687 |
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-ral 3060 df-rex 3069 df-rab 3434 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 df-br 5149 df-opab 5211 df-xp 5695 |
This theorem is referenced by: fpwwe 10684 efgrelexlema 19782 brsslt 27845 rgrprop 29593 rusgrprop 29595 bropabg 43313 clcllaw 48035 asslawass 48037 |
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