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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 builder 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 4584 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | bropaex12.1 | . . . . 5 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓} | |
3 | 2 | eleq2i 2680 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
4 | 1, 3 | bitri 263 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
5 | elopaelxp 5114 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
6 | 4, 5 | sylbi 206 | . 2 ⊢ (𝐴𝑅𝐵 → 〈𝐴, 𝐵〉 ∈ (V × V)) |
7 | opelxp 5070 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
8 | 6, 7 | sylib 207 | 1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 {copab 4642 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 |
This theorem is referenced by: fpwwe 9347 efgrelexlema 17985 clcllaw 41617 asslawass 41619 |
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