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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 3739 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.) |
Ref | Expression |
---|---|
pw1dom2 | ⊢ 2o ≼ 𝒫 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nep0 4097 | . . . 4 ⊢ ∅ ≠ {∅} | |
2 | 0ex 4063 | . . . . 5 ⊢ ∅ ∈ V | |
3 | p0ex 4120 | . . . . 5 ⊢ {∅} ∈ V | |
4 | pr2ne 7065 | . . . . 5 ⊢ ((∅ ∈ V ∧ {∅} ∈ V) → ({∅, {∅}} ≈ 2o ↔ ∅ ≠ {∅})) | |
5 | 2, 3, 4 | mp2an 423 | . . . 4 ⊢ ({∅, {∅}} ≈ 2o ↔ ∅ ≠ {∅}) |
6 | 1, 5 | mpbir 145 | . . 3 ⊢ {∅, {∅}} ≈ 2o |
7 | 6 | ensymi 6684 | . 2 ⊢ 2o ≈ {∅, {∅}} |
8 | 3 | pwex 4115 | . . . 4 ⊢ 𝒫 {∅} ∈ V |
9 | pwpw0ss 3739 | . . . 4 ⊢ {∅, {∅}} ⊆ 𝒫 {∅} | |
10 | ssdomg 6680 | . . . 4 ⊢ (𝒫 {∅} ∈ V → ({∅, {∅}} ⊆ 𝒫 {∅} → {∅, {∅}} ≼ 𝒫 {∅})) | |
11 | 8, 9, 10 | mp2 16 | . . 3 ⊢ {∅, {∅}} ≼ 𝒫 {∅} |
12 | df1o2 6334 | . . . 4 ⊢ 1o = {∅} | |
13 | 12 | pweqi 3519 | . . 3 ⊢ 𝒫 1o = 𝒫 {∅} |
14 | 11, 13 | breqtrri 3963 | . 2 ⊢ {∅, {∅}} ≼ 𝒫 1o |
15 | endomtr 6692 | . 2 ⊢ ((2o ≈ {∅, {∅}} ∧ {∅, {∅}} ≼ 𝒫 1o) → 2o ≼ 𝒫 1o) | |
16 | 7, 14, 15 | mp2an 423 | 1 ⊢ 2o ≼ 𝒫 1o |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1481 ≠ wne 2309 Vcvv 2689 ⊆ wss 3076 ∅c0 3368 𝒫 cpw 3515 {csn 3532 {cpr 3533 class class class wbr 3937 1oc1o 6314 2oc2o 6315 ≈ cen 6640 ≼ cdom 6641 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1o 6321 df-2o 6322 df-er 6437 df-en 6643 df-dom 6644 |
This theorem is referenced by: pwf1oexmid 13367 |
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