Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4584 | . 2 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | opprc 4362 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈𝑋, 𝑌〉 = ∅) | |
3 | 0neqopab 6596 | . . . . 5 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | eleq1 2676 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 = ∅ → (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
5 | 3, 4 | mtbiri 316 | . . . 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 112 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
8 | 1, 7 | sylbi 206 | 1 ⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 class class class wbr 4583 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 |
This theorem is referenced by: brfvopab 6598 bropopvvv 7142 bropfvvvvlem 7143 isfunc 16347 eqgval 17466 rgrprop 40760 rusgrprop 40762 wlkbProp 40817 |
Copyright terms: Public domain | W3C validator |