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Theorem pw1on 7219
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 6424 . . . . . 6 1o = {∅}
2 elsni 3609 . . . . . . . 8 (𝑥 ∈ {∅} → 𝑥 = ∅)
3 0elpw 4161 . . . . . . . 8 ∅ ∈ 𝒫 1o
42, 3eqeltrdi 2268 . . . . . . 7 (𝑥 ∈ {∅} → 𝑥 ∈ 𝒫 1o)
54ssriv 3159 . . . . . 6 {∅} ⊆ 𝒫 1o
61, 5eqsstri 3187 . . . . 5 1o ⊆ 𝒫 1o
7 sspwb 4213 . . . . 5 (1o ⊆ 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
86, 7mpbi 145 . . . 4 𝒫 1o ⊆ 𝒫 𝒫 1o
9 dftr4 4103 . . . 4 (Tr 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
108, 9mpbir 146 . . 3 Tr 𝒫 1o
11 elpwi 3583 . . . . . . . . 9 (𝑥 ∈ 𝒫 1o𝑥 ⊆ 1o)
1211sselda 3155 . . . . . . . 8 ((𝑥 ∈ 𝒫 1o𝑦𝑥) → 𝑦 ∈ 1o)
13 el1o 6432 . . . . . . . 8 (𝑦 ∈ 1o𝑦 = ∅)
1412, 13sylib 122 . . . . . . 7 ((𝑥 ∈ 𝒫 1o𝑦𝑥) → 𝑦 = ∅)
15 0ss 3461 . . . . . . 7 ∅ ⊆ 𝑥
1614, 15eqsstrdi 3207 . . . . . 6 ((𝑥 ∈ 𝒫 1o𝑦𝑥) → 𝑦𝑥)
1716ralrimiva 2550 . . . . 5 (𝑥 ∈ 𝒫 1o → ∀𝑦𝑥 𝑦𝑥)
18 dftr3 4102 . . . . 5 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
1917, 18sylibr 134 . . . 4 (𝑥 ∈ 𝒫 1o → Tr 𝑥)
2019rgen 2530 . . 3 𝑥 ∈ 𝒫 1oTr 𝑥
21 dford3 4364 . . 3 (Ord 𝒫 1o ↔ (Tr 𝒫 1o ∧ ∀𝑥 ∈ 𝒫 1oTr 𝑥))
2210, 20, 21mpbir2an 942 . 2 Ord 𝒫 1o
23 1oex 6419 . . 3 1o ∈ V
2423pwex 4180 . 2 𝒫 1o ∈ V
25 elon2 4373 . 2 (𝒫 1o ∈ On ↔ (Ord 𝒫 1o ∧ 𝒫 1o ∈ V))
2622, 24, 25mpbir2an 942 1 𝒫 1o ∈ On
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  wral 2455  Vcvv 2737  wss 3129  c0 3422  𝒫 cpw 3574  {csn 3591  Tr wtr 4098  Ord word 4359  Oncon0 4360  1oc1o 6404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-tr 4099  df-iord 4363  df-on 4365  df-suc 4368  df-1o 6411
This theorem is referenced by:  pw1ne1  7222  sucpw1nss3  7228  onntri35  7230  onntri45  7234
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