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Mirrors > Home > ILE Home > Th. List > Mathboxes > pw1dom2 | GIF version |
Description: The power set of 1o dominates 2o. Also see pwpw0ss 3731 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.) |
Ref | Expression |
---|---|
pw1dom2 | ⊢ 2o ≼ 𝒫 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nep0 4089 | . . . 4 ⊢ ∅ ≠ {∅} | |
2 | 0ex 4055 | . . . . 5 ⊢ ∅ ∈ V | |
3 | p0ex 4112 | . . . . 5 ⊢ {∅} ∈ V | |
4 | pr2ne 7048 | . . . . 5 ⊢ ((∅ ∈ V ∧ {∅} ∈ V) → ({∅, {∅}} ≈ 2o ↔ ∅ ≠ {∅})) | |
5 | 2, 3, 4 | mp2an 422 | . . . 4 ⊢ ({∅, {∅}} ≈ 2o ↔ ∅ ≠ {∅}) |
6 | 1, 5 | mpbir 145 | . . 3 ⊢ {∅, {∅}} ≈ 2o |
7 | 6 | ensymi 6676 | . 2 ⊢ 2o ≈ {∅, {∅}} |
8 | 3 | pwex 4107 | . . . 4 ⊢ 𝒫 {∅} ∈ V |
9 | pwpw0ss 3731 | . . . 4 ⊢ {∅, {∅}} ⊆ 𝒫 {∅} | |
10 | ssdomg 6672 | . . . 4 ⊢ (𝒫 {∅} ∈ V → ({∅, {∅}} ⊆ 𝒫 {∅} → {∅, {∅}} ≼ 𝒫 {∅})) | |
11 | 8, 9, 10 | mp2 16 | . . 3 ⊢ {∅, {∅}} ≼ 𝒫 {∅} |
12 | df1o2 6326 | . . . 4 ⊢ 1o = {∅} | |
13 | 12 | pweqi 3514 | . . 3 ⊢ 𝒫 1o = 𝒫 {∅} |
14 | 11, 13 | breqtrri 3955 | . 2 ⊢ {∅, {∅}} ≼ 𝒫 1o |
15 | endomtr 6684 | . 2 ⊢ ((2o ≈ {∅, {∅}} ∧ {∅, {∅}} ≼ 𝒫 1o) → 2o ≼ 𝒫 1o) | |
16 | 7, 14, 15 | mp2an 422 | 1 ⊢ 2o ≼ 𝒫 1o |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 ≠ wne 2308 Vcvv 2686 ⊆ wss 3071 ∅c0 3363 𝒫 cpw 3510 {csn 3527 {cpr 3528 class class class wbr 3929 1oc1o 6306 2oc2o 6307 ≈ cen 6632 ≼ cdom 6633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1o 6313 df-2o 6314 df-er 6429 df-en 6635 df-dom 6636 |
This theorem is referenced by: pwf1oexmid 13194 |
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