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Mirrors > Home > NFE Home > Th. List > elunii | GIF version |
Description: Membership in class union. (Contributed by NM, 24-Mar-1995.) |
Ref | Expression |
---|---|
elunii | ⊢ ((A ∈ B ∧ B ∈ C) → A ∈ ∪C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2414 | . . . . 5 ⊢ (x = B → (A ∈ x ↔ A ∈ B)) | |
2 | eleq1 2413 | . . . . 5 ⊢ (x = B → (x ∈ C ↔ B ∈ C)) | |
3 | 1, 2 | anbi12d 691 | . . . 4 ⊢ (x = B → ((A ∈ x ∧ x ∈ C) ↔ (A ∈ B ∧ B ∈ C))) |
4 | 3 | spcegv 2940 | . . 3 ⊢ (B ∈ C → ((A ∈ B ∧ B ∈ C) → ∃x(A ∈ x ∧ x ∈ C))) |
5 | 4 | anabsi7 792 | . 2 ⊢ ((A ∈ B ∧ B ∈ C) → ∃x(A ∈ x ∧ x ∈ C)) |
6 | eluni 3894 | . 2 ⊢ (A ∈ ∪C ↔ ∃x(A ∈ x ∧ x ∈ C)) | |
7 | 5, 6 | sylibr 203 | 1 ⊢ ((A ∈ B ∧ B ∈ C) → A ∈ ∪C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∪cuni 3891 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-uni 3892 |
This theorem is referenced by: ssuni 3913 unipw 4117 nnadjoin 4520 sfinltfin 4535 nulnnn 4556 |
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