New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > pw1un | GIF version |
Description: Unit power class distributes over union. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
pw1un | ⊢ ℘1(A ∪ B) = (℘1A ∪ ℘1B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexun 3443 | . . 3 ⊢ (∃y ∈ (A ∪ B)x = {y} ↔ (∃y ∈ A x = {y} ∨ ∃y ∈ B x = {y})) | |
2 | elpw1 4144 | . . 3 ⊢ (x ∈ ℘1(A ∪ B) ↔ ∃y ∈ (A ∪ B)x = {y}) | |
3 | elun 3220 | . . . 4 ⊢ (x ∈ (℘1A ∪ ℘1B) ↔ (x ∈ ℘1A ∨ x ∈ ℘1B)) | |
4 | elpw1 4144 | . . . . 5 ⊢ (x ∈ ℘1A ↔ ∃y ∈ A x = {y}) | |
5 | elpw1 4144 | . . . . 5 ⊢ (x ∈ ℘1B ↔ ∃y ∈ B x = {y}) | |
6 | 4, 5 | orbi12i 507 | . . . 4 ⊢ ((x ∈ ℘1A ∨ x ∈ ℘1B) ↔ (∃y ∈ A x = {y} ∨ ∃y ∈ B x = {y})) |
7 | 3, 6 | bitri 240 | . . 3 ⊢ (x ∈ (℘1A ∪ ℘1B) ↔ (∃y ∈ A x = {y} ∨ ∃y ∈ B x = {y})) |
8 | 1, 2, 7 | 3bitr4i 268 | . 2 ⊢ (x ∈ ℘1(A ∪ B) ↔ x ∈ (℘1A ∪ ℘1B)) |
9 | 8 | eqriv 2350 | 1 ⊢ ℘1(A ∪ B) = (℘1A ∪ ℘1B) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 357 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∪ cun 3207 {csn 3737 ℘1cpw1 4135 |
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 ax-nin 4078 ax-sn 4087 |
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-ne 2518 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-1c 4136 df-pw1 4137 |
This theorem is referenced by: pw1equn 4331 pw1eqadj 4332 ncfinraise 4481 tfindi 4496 tfinsuc 4498 sfindbl 4530 tcdi 6164 ce0addcnnul 6179 ce2 6192 |
Copyright terms: Public domain | W3C validator |