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| Mirrors > Home > MPE Home > Th. List > brabv | Structured version Visualization version GIF version | ||
| Description: If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
| Ref | Expression |
|---|---|
| brabv | ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-br 5144 | . 2 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
| 2 | opprc 4896 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
| 3 | 0nelopab 5572 | . . . . 5 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 4 | eleq1 2829 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 = ∅ → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
| 5 | 3, 4 | mtbiri 327 | . . . 4 ⊢ (〈𝑋, 𝑌〉 = ∅ → ¬ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
| 6 | 2, 5 | syl 17 | . . 3 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ¬ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
| 7 | 6 | con4i 114 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
| 8 | 1, 7 | sylbi 217 | 1 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 class class class wbr 5143 {copab 5205 |
| 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-ne 2941 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 |
| This theorem is referenced by: brfvopab 7490 bropopvvv 8115 bropfvvvvlem 8116 isfunc 17909 eqgval 19195 bj-imdirval3 37185 upwlkbprop 48054 |
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