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Theorem hashinfuni 10530
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 4507 . . . . . 6 ω ∈ V
21snid 3556 . . . . 5 ω ∈ {ω}
3 elun2 3244 . . . . 5 (ω ∈ {ω} → ω ∈ (ω ∪ {ω}))
4 breq1 3932 . . . . . 6 (𝑦 = ω → (𝑦𝐴 ↔ ω ≼ 𝐴))
54elrab3 2841 . . . . 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 2837 . . . . . . 7 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → 𝑧 ∈ (ω ∪ {ω}))
9 elun 3217 . . . . . . 7 (𝑧 ∈ (ω ∪ {ω}) ↔ (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
108, 9sylib 121 . . . . . 6 (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} → (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
11 ordom 4520 . . . . . . . 8 Ord ω
12 ordelss 4301 . . . . . . . 8 ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω)
1311, 12mpan 420 . . . . . . 7 (𝑧 ∈ ω → 𝑧 ⊆ ω)
14 elsni 3545 . . . . . . . 8 (𝑧 ∈ {ω} → 𝑧 = ω)
15 eqimss 3151 . . . . . . . 8 (𝑧 = ω → 𝑧 ⊆ ω)
1614, 15syl 14 . . . . . . 7 (𝑧 ∈ {ω} → 𝑧 ⊆ ω)
1713, 16jaoi 705 . . . . . 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 2505 . . 3 (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}𝑧 ⊆ ω)
21 ssunieq 3769 . . 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 2145 1 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 697   = wceq 1331  wcel 1480  wral 2416  {crab 2420  cun 3069  wss 3071  {csn 3527   cuni 3736   class class class wbr 3929  Ord word 4284  ωcom 4504  cdom 6633
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-tr 4027  df-iord 4288  df-suc 4293  df-iom 4505
This theorem is referenced by:  hashinfom  10531
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