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Mirrors > Home > ILE Home > Th. List > pw1dom2 | GIF version |
Description: The power set of 1o dominates 2o. Also see pwpw0ss 3791 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.) |
Ref | Expression |
---|---|
pw1dom2 | ⊢ 2o ≼ 𝒫 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nep0 4151 | . . . 4 ⊢ ∅ ≠ {∅} | |
2 | 0ex 4116 | . . . . 5 ⊢ ∅ ∈ V | |
3 | p0ex 4174 | . . . . 5 ⊢ {∅} ∈ V | |
4 | pr2ne 7169 | . . . . 5 ⊢ ((∅ ∈ V ∧ {∅} ∈ V) → ({∅, {∅}} ≈ 2o ↔ ∅ ≠ {∅})) | |
5 | 2, 3, 4 | mp2an 424 | . . . 4 ⊢ ({∅, {∅}} ≈ 2o ↔ ∅ ≠ {∅}) |
6 | 1, 5 | mpbir 145 | . . 3 ⊢ {∅, {∅}} ≈ 2o |
7 | 6 | ensymi 6760 | . 2 ⊢ 2o ≈ {∅, {∅}} |
8 | 3 | pwex 4169 | . . . 4 ⊢ 𝒫 {∅} ∈ V |
9 | pwpw0ss 3791 | . . . 4 ⊢ {∅, {∅}} ⊆ 𝒫 {∅} | |
10 | ssdomg 6756 | . . . 4 ⊢ (𝒫 {∅} ∈ V → ({∅, {∅}} ⊆ 𝒫 {∅} → {∅, {∅}} ≼ 𝒫 {∅})) | |
11 | 8, 9, 10 | mp2 16 | . . 3 ⊢ {∅, {∅}} ≼ 𝒫 {∅} |
12 | df1o2 6408 | . . . 4 ⊢ 1o = {∅} | |
13 | 12 | pweqi 3570 | . . 3 ⊢ 𝒫 1o = 𝒫 {∅} |
14 | 11, 13 | breqtrri 4016 | . 2 ⊢ {∅, {∅}} ≼ 𝒫 1o |
15 | endomtr 6768 | . 2 ⊢ ((2o ≈ {∅, {∅}} ∧ {∅, {∅}} ≼ 𝒫 1o) → 2o ≼ 𝒫 1o) | |
16 | 7, 14, 15 | mp2an 424 | 1 ⊢ 2o ≼ 𝒫 1o |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2141 ≠ wne 2340 Vcvv 2730 ⊆ wss 3121 ∅c0 3414 𝒫 cpw 3566 {csn 3583 {cpr 3584 class class class wbr 3989 1oc1o 6388 2oc2o 6389 ≈ cen 6716 ≼ cdom 6717 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 df-dom 6720 |
This theorem is referenced by: pwf1oexmid 14032 |
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