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Mirrors > Home > ILE Home > Th. List > pwprss | GIF version |
Description: The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.) |
Ref | Expression |
---|---|
pwprss | ⊢ ({∅, {A}} ∪ {{B}, {A, B}}) ⊆ 𝒫 {A, B} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 | . . . . . 6 ⊢ x ∈ V | |
2 | 1 | elpr 3385 | . . . . 5 ⊢ (x ∈ {∅, {A}} ↔ (x = ∅ ∨ x = {A})) |
3 | 1 | elpr 3385 | . . . . 5 ⊢ (x ∈ {{B}, {A, B}} ↔ (x = {B} ∨ x = {A, B})) |
4 | 2, 3 | orbi12i 680 | . . . 4 ⊢ ((x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}}) ↔ ((x = ∅ ∨ x = {A}) ∨ (x = {B} ∨ x = {A, B}))) |
5 | ssprr 3518 | . . . 4 ⊢ (((x = ∅ ∨ x = {A}) ∨ (x = {B} ∨ x = {A, B})) → x ⊆ {A, B}) | |
6 | 4, 5 | sylbi 114 | . . 3 ⊢ ((x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}}) → x ⊆ {A, B}) |
7 | elun 3078 | . . 3 ⊢ (x ∈ ({∅, {A}} ∪ {{B}, {A, B}}) ↔ (x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}})) | |
8 | 1 | elpw 3357 | . . 3 ⊢ (x ∈ 𝒫 {A, B} ↔ x ⊆ {A, B}) |
9 | 6, 7, 8 | 3imtr4i 190 | . 2 ⊢ (x ∈ ({∅, {A}} ∪ {{B}, {A, B}}) → x ∈ 𝒫 {A, B}) |
10 | 9 | ssriv 2943 | 1 ⊢ ({∅, {A}} ∪ {{B}, {A, B}}) ⊆ 𝒫 {A, B} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 628 = wceq 1242 ∈ wcel 1390 ∪ cun 2909 ⊆ wss 2911 ∅c0 3218 𝒫 cpw 3351 {csn 3367 {cpr 3368 |
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-in1 544 ax-in2 545 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-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
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-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 |
This theorem is referenced by: pwpwpw0ss 3569 ord3ex 3932 |
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