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Theorem hashinfuni 10491
Description: The ordinal size of an infinite set is ω. (Contributed by Jim Kingdon, 20-Feb-2022.)
Assertion
Ref Expression
hashinfuni (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
Distinct variable group:   𝑦,𝐴

Proof of Theorem hashinfuni
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 omex 4477 . . . . . 6 ω ∈ V
21snid 3526 . . . . 5 ω ∈ {ω}
3 elun2 3214 . . . . 5 (ω ∈ {ω} → ω ∈ (ω ∪ {ω}))
4 breq1 3902 . . . . . 6 (𝑦 = ω → (𝑦𝐴 ↔ ω ≼ 𝐴))
54elrab3 2814 . . . . 5 (ω ∈ (ω ∪ {ω}) → (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ↔ ω ≼ 𝐴))
62, 3, 5mp2b 8 . . . 4 (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ↔ ω ≼ 𝐴)
76biimpri 132 . . 3 (ω ≼ 𝐴 → ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
8 elrabi 2810 . . . . . . 7 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → 𝑧 ∈ (ω ∪ {ω}))
9 elun 3187 . . . . . . 7 (𝑧 ∈ (ω ∪ {ω}) ↔ (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
108, 9sylib 121 . . . . . 6 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
11 ordom 4490 . . . . . . . 8 Ord ω
12 ordelss 4271 . . . . . . . 8 ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω)
1311, 12mpan 420 . . . . . . 7 (𝑧 ∈ ω → 𝑧 ⊆ ω)
14 elsni 3515 . . . . . . . 8 (𝑧 ∈ {ω} → 𝑧 = ω)
15 eqimss 3121 . . . . . . . 8 (𝑧 = ω → 𝑧 ⊆ ω)
1614, 15syl 14 . . . . . . 7 (𝑧 ∈ {ω} → 𝑧 ⊆ ω)
1713, 16jaoi 690 . . . . . 6 ((𝑧 ∈ ω ∨ 𝑧 ∈ {ω}) → 𝑧 ⊆ ω)
1810, 17syl 14 . . . . 5 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → 𝑧 ⊆ ω)
1918adantl 275 . . . 4 ((ω ≼ 𝐴𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}) → 𝑧 ⊆ ω)
2019ralrimiva 2482 . . 3 (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}𝑧 ⊆ ω)
21 ssunieq 3739 . . 3 ((ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∧ ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}𝑧 ⊆ ω) → ω = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
227, 20, 21syl2anc 408 . 2 (ω ≼ 𝐴 → ω = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
2322eqcomd 2123 1 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 682   = wceq 1316  wcel 1465  wral 2393  {crab 2397  cun 3039  wss 3041  {csn 3497   cuni 3706   class class class wbr 3899  Ord word 4254  ωcom 4474  cdom 6601
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-tr 3997  df-iord 4258  df-suc 4263  df-iom 4475
This theorem is referenced by:  hashinfom  10492
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