Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  pw1on GIF version

Theorem pw1on 7144
 Description: The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
Assertion
Ref Expression
pw1on 𝒫 1o ∈ On

Proof of Theorem pw1on
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df1o2 6370 . . . . . 6 1o = {∅}
2 elsni 3578 . . . . . . . 8 (𝑥 ∈ {∅} → 𝑥 = ∅)
3 0elpw 4124 . . . . . . . 8 ∅ ∈ 𝒫 1o
42, 3eqeltrdi 2248 . . . . . . 7 (𝑥 ∈ {∅} → 𝑥 ∈ 𝒫 1o)
54ssriv 3132 . . . . . 6 {∅} ⊆ 𝒫 1o
61, 5eqsstri 3160 . . . . 5 1o ⊆ 𝒫 1o
7 sspwb 4175 . . . . 5 (1o ⊆ 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
86, 7mpbi 144 . . . 4 𝒫 1o ⊆ 𝒫 𝒫 1o
9 dftr4 4067 . . . 4 (Tr 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
108, 9mpbir 145 . . 3 Tr 𝒫 1o
11 elpwi 3552 . . . . . . . . 9 (𝑥 ∈ 𝒫 1o𝑥 ⊆ 1o)
1211sselda 3128 . . . . . . . 8 ((𝑥 ∈ 𝒫 1o𝑦𝑥) → 𝑦 ∈ 1o)
13 el1o 6378 . . . . . . . 8 (𝑦 ∈ 1o𝑦 = ∅)
1412, 13sylib 121 . . . . . . 7 ((𝑥 ∈ 𝒫 1o𝑦𝑥) → 𝑦 = ∅)
15 0ss 3432 . . . . . . 7 ∅ ⊆ 𝑥
1614, 15eqsstrdi 3180 . . . . . 6 ((𝑥 ∈ 𝒫 1o𝑦𝑥) → 𝑦𝑥)
1716ralrimiva 2530 . . . . 5 (𝑥 ∈ 𝒫 1o → ∀𝑦𝑥 𝑦𝑥)
18 dftr3 4066 . . . . 5 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
1917, 18sylibr 133 . . . 4 (𝑥 ∈ 𝒫 1o → Tr 𝑥)
2019rgen 2510 . . 3 𝑥 ∈ 𝒫 1oTr 𝑥
21 dford3 4326 . . 3 (Ord 𝒫 1o ↔ (Tr 𝒫 1o ∧ ∀𝑥 ∈ 𝒫 1oTr 𝑥))
2210, 20, 21mpbir2an 927 . 2 Ord 𝒫 1o
23 1oex 6365 . . 3 1o ∈ V
2423pwex 4143 . 2 𝒫 1o ∈ V
25 elon2 4335 . 2 (𝒫 1o ∈ On ↔ (Ord 𝒫 1o ∧ 𝒫 1o ∈ V))
2622, 24, 25mpbir2an 927 1 𝒫 1o ∈ On
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1335   ∈ wcel 2128  ∀wral 2435  Vcvv 2712   ⊆ wss 3102  ∅c0 3394  𝒫 cpw 3543  {csn 3560  Tr wtr 4062  Ord word 4321  Oncon0 4322  1oc1o 6350 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4325  df-on 4327  df-suc 4330  df-1o 6357 This theorem is referenced by:  pw1ne1  7147  sucpw1nss3  7153  onntri35  7155  onntri45  7159
 Copyright terms: Public domain W3C validator