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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 5069 | . 2 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | opprc 4828 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
3 | 0nelopab 5454 | . . . . 5 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | eleq1 2902 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 = ∅ → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
5 | 3, 4 | mtbiri 329 | . . . 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 219 | 1 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 〈cop 4575 class class class wbr 5068 {copab 5130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 |
This theorem is referenced by: brfvopab 7213 bropopvvv 7787 bropfvvvvlem 7788 isfunc 17136 eqgval 18331 bj-imdirval3 34476 upwlkbprop 44020 |
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