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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 4788 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | bropaex12.1 | . . . . 5 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓} | |
3 | 2 | eleq2i 2842 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
4 | 1, 3 | bitri 264 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
5 | elopaelxp 5330 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
6 | 4, 5 | sylbi 207 | . 2 ⊢ (𝐴𝑅𝐵 → 〈𝐴, 𝐵〉 ∈ (V × V)) |
7 | opelxp 5285 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
8 | 6, 7 | sylib 208 | 1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 〈cop 4323 class class class wbr 4787 {copab 4847 × cxp 5248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-br 4788 df-opab 4848 df-xp 5256 |
This theorem is referenced by: fpwwe 9674 efgrelexlema 18369 brsslt 32237 clcllaw 42352 asslawass 42354 |
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