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Mirrors > Home > NFE Home > Th. List > iundif2 | GIF version |
Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4020 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.) |
Ref | Expression |
---|---|
iundif2 | ⊢ ∪x ∈ A (B ∖ C) = (B ∖ ∩x ∈ A C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3221 | . . . . 5 ⊢ (y ∈ (B ∖ C) ↔ (y ∈ B ∧ ¬ y ∈ C)) | |
2 | 1 | rexbii 2639 | . . . 4 ⊢ (∃x ∈ A y ∈ (B ∖ C) ↔ ∃x ∈ A (y ∈ B ∧ ¬ y ∈ C)) |
3 | r19.42v 2765 | . . . 4 ⊢ (∃x ∈ A (y ∈ B ∧ ¬ y ∈ C) ↔ (y ∈ B ∧ ∃x ∈ A ¬ y ∈ C)) | |
4 | rexnal 2625 | . . . . . 6 ⊢ (∃x ∈ A ¬ y ∈ C ↔ ¬ ∀x ∈ A y ∈ C) | |
5 | vex 2862 | . . . . . . 7 ⊢ y ∈ V | |
6 | eliin 3974 | . . . . . . 7 ⊢ (y ∈ V → (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C) |
8 | 4, 7 | xchbinxr 302 | . . . . 5 ⊢ (∃x ∈ A ¬ y ∈ C ↔ ¬ y ∈ ∩x ∈ A C) |
9 | 8 | anbi2i 675 | . . . 4 ⊢ ((y ∈ B ∧ ∃x ∈ A ¬ y ∈ C) ↔ (y ∈ B ∧ ¬ y ∈ ∩x ∈ A C)) |
10 | 2, 3, 9 | 3bitri 262 | . . 3 ⊢ (∃x ∈ A y ∈ (B ∖ C) ↔ (y ∈ B ∧ ¬ y ∈ ∩x ∈ A C)) |
11 | eliun 3973 | . . 3 ⊢ (y ∈ ∪x ∈ A (B ∖ C) ↔ ∃x ∈ A y ∈ (B ∖ C)) | |
12 | eldif 3221 | . . 3 ⊢ (y ∈ (B ∖ ∩x ∈ A C) ↔ (y ∈ B ∧ ¬ y ∈ ∩x ∈ A C)) | |
13 | 10, 11, 12 | 3bitr4i 268 | . 2 ⊢ (y ∈ ∪x ∈ A (B ∖ C) ↔ y ∈ (B ∖ ∩x ∈ A C)) |
14 | 13 | eqriv 2350 | 1 ⊢ ∪x ∈ A (B ∖ C) = (B ∖ ∩x ∈ A C) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 Vcvv 2859 ∖ cdif 3206 ∪ciun 3969 ∩ciin 3970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-nan 1288 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-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-iun 3971 df-iin 3972 |
This theorem is referenced by: (None) |
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