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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 5031 | . 2 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | opprc 4788 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
3 | 0nelopab 5417 | . . . . 5 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | eleq1 2877 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 = ∅ → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
5 | 3, 4 | mtbiri 330 | . . . 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 220 | 1 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 class class class wbr 5030 {copab 5092 |
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-ne 2988 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 |
This theorem is referenced by: brfvopab 7190 bropopvvv 7768 bropfvvvvlem 7769 isfunc 17126 eqgval 18321 bj-imdirval3 34599 upwlkbprop 44366 |
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