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Theorem hashinfuni 10774
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 4606 . . . . . 6 ω ∈ V
21snid 3637 . . . . 5 ω ∈ {ω}
3 elun2 3317 . . . . 5 (ω ∈ {ω} → ω ∈ (ω ∪ {ω}))
4 breq1 4020 . . . . . 6 (𝑦 = ω → (𝑦𝐴 ↔ ω ≼ 𝐴))
54elrab3 2908 . . . . 5 (ω ∈ (ω ∪ {ω}) → (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ↔ ω ≼ 𝐴))
62, 3, 5mp2b 8 . . . 4 (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ↔ ω ≼ 𝐴)
76biimpri 133 . . 3 (ω ≼ 𝐴 → ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
8 elrabi 2904 . . . . . . 7 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → 𝑧 ∈ (ω ∪ {ω}))
9 elun 3290 . . . . . . 7 (𝑧 ∈ (ω ∪ {ω}) ↔ (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
108, 9sylib 122 . . . . . 6 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
11 ordom 4620 . . . . . . . 8 Ord ω
12 ordelss 4393 . . . . . . . 8 ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω)
1311, 12mpan 424 . . . . . . 7 (𝑧 ∈ ω → 𝑧 ⊆ ω)
14 elsni 3624 . . . . . . . 8 (𝑧 ∈ {ω} → 𝑧 = ω)
15 eqimss 3223 . . . . . . . 8 (𝑧 = ω → 𝑧 ⊆ ω)
1614, 15syl 14 . . . . . . 7 (𝑧 ∈ {ω} → 𝑧 ⊆ ω)
1713, 16jaoi 717 . . . . . 6 ((𝑧 ∈ ω ∨ 𝑧 ∈ {ω}) → 𝑧 ⊆ ω)
1810, 17syl 14 . . . . 5 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → 𝑧 ⊆ ω)
1918adantl 277 . . . 4 ((ω ≼ 𝐴𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}) → 𝑧 ⊆ ω)
2019ralrimiva 2562 . . 3 (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}𝑧 ⊆ ω)
21 ssunieq 3856 . . 3 ((ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∧ ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}𝑧 ⊆ ω) → ω = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
227, 20, 21syl2anc 411 . 2 (ω ≼ 𝐴 → ω = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
2322eqcomd 2194 1 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 709   = wceq 1363  wcel 2159  wral 2467  {crab 2471  cun 3141  wss 3143  {csn 3606   cuni 3823   class class class wbr 4017  Ord word 4376  ωcom 4603  cdom 6756
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-iinf 4601
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-br 4018  df-tr 4116  df-iord 4380  df-suc 4385  df-iom 4604
This theorem is referenced by:  hashinfom  10775
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