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Theorem hashinfuni 10020
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 4371 . . . . . 6 ω ∈ V
21snid 3449 . . . . 5 ω ∈ {ω}
3 elun2 3152 . . . . 5 (ω ∈ {ω} → ω ∈ (ω ∪ {ω}))
4 breq1 3814 . . . . . 6 (𝑦 = ω → (𝑦𝐴 ↔ ω ≼ 𝐴))
54elrab3 2760 . . . . 5 (ω ∈ (ω ∪ {ω}) → (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ↔ ω ≼ 𝐴))
62, 3, 5mp2b 8 . . . 4 (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ↔ ω ≼ 𝐴)
76biimpri 131 . . 3 (ω ≼ 𝐴 → ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
8 elrabi 2756 . . . . . . 7 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → 𝑧 ∈ (ω ∪ {ω}))
9 elun 3125 . . . . . . 7 (𝑧 ∈ (ω ∪ {ω}) ↔ (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
108, 9sylib 120 . . . . . 6 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
11 ordom 4384 . . . . . . . 8 Ord ω
12 ordelss 4170 . . . . . . . 8 ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω)
1311, 12mpan 415 . . . . . . 7 (𝑧 ∈ ω → 𝑧 ⊆ ω)
14 elsni 3440 . . . . . . . 8 (𝑧 ∈ {ω} → 𝑧 = ω)
15 eqimss 3062 . . . . . . . 8 (𝑧 = ω → 𝑧 ⊆ ω)
1614, 15syl 14 . . . . . . 7 (𝑧 ∈ {ω} → 𝑧 ⊆ ω)
1713, 16jaoi 669 . . . . . 6 ((𝑧 ∈ ω ∨ 𝑧 ∈ {ω}) → 𝑧 ⊆ ω)
1810, 17syl 14 . . . . 5 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → 𝑧 ⊆ ω)
1918adantl 271 . . . 4 ((ω ≼ 𝐴𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}) → 𝑧 ⊆ ω)
2019ralrimiva 2440 . . 3 (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}𝑧 ⊆ ω)
21 ssunieq 3660 . . 3 ((ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∧ ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}𝑧 ⊆ ω) → ω = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
227, 20, 21syl2anc 403 . 2 (ω ≼ 𝐴 → ω = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
2322eqcomd 2088 1 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 662   = wceq 1285  wcel 1434  wral 2353  {crab 2357  cun 2982  wss 2984  {csn 3422   cuni 3627   class class class wbr 3811  Ord word 4153  ωcom 4368  cdom 6386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-tr 3902  df-iord 4157  df-suc 4162  df-iom 4369
This theorem is referenced by:  hashinfom  10021
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