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Mirrors > Home > ILE Home > Th. List > uniexg | GIF version |
Description: The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent A ∈ 𝑉 instead of A ∈ V to make the theorem more general and thus shorten some proofs; obviously the universal class constant V is one possible substitution for class variable 𝑉. (Contributed by NM, 25-Nov-1994.) |
Ref | Expression |
---|---|
uniexg | ⊢ (A ∈ 𝑉 → ∪ A ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3580 | . . 3 ⊢ (x = A → ∪ x = ∪ A) | |
2 | 1 | eleq1d 2103 | . 2 ⊢ (x = A → (∪ x ∈ V ↔ ∪ A ∈ V)) |
3 | vex 2554 | . . 3 ⊢ x ∈ V | |
4 | 3 | uniex 4140 | . 2 ⊢ ∪ x ∈ V |
5 | 2, 4 | vtoclg 2607 | 1 ⊢ (A ∈ 𝑉 → ∪ A ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ∪ cuni 3571 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-uni 3572 |
This theorem is referenced by: snnex 4147 uniexb 4171 ssonuni 4180 dmexg 4539 rnexg 4540 elxp4 4751 elxp5 4752 relrnfvex 5136 fvexg 5137 sefvex 5139 riotaexg 5415 iunexg 5688 1stvalg 5711 2ndvalg 5712 cnvf1o 5788 brtpos2 5807 tfrlemiex 5886 en1bg 6216 en1uniel 6220 |
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