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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 5031 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | bropaex12.1 | . . . . 5 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓} | |
3 | 2 | eleq2i 2881 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
4 | 1, 3 | bitri 278 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓}) |
5 | elopaelxp 5605 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
6 | 4, 5 | sylbi 220 | . 2 ⊢ (𝐴𝑅𝐵 → 〈𝐴, 𝐵〉 ∈ (V × V)) |
7 | opelxp 5555 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
8 | 6, 7 | sylib 221 | 1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 {copab 5092 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 |
This theorem is referenced by: fpwwe 10057 efgrelexlema 18867 rgrprop 27350 rusgrprop 27352 brsslt 33367 clcllaw 44451 asslawass 44453 |
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