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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 5149 | . 2 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | opprc 4901 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
3 | 0nelopab 5577 | . . . . 5 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | eleq1 2827 | . . . . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 〈cop 4637 class class class wbr 5148 {copab 5210 |
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-ne 2939 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 |
This theorem is referenced by: brfvopab 7490 bropopvvv 8114 bropfvvvvlem 8115 isfunc 17915 eqgval 19208 bj-imdirval3 37167 upwlkbprop 47982 |
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