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Theorem elharval 9009
Description: The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9007, this implies that the Hartogs number of a set is greater than all ordinals that set dominates. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
elharval (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌𝑋))

Proof of Theorem elharval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elfvex 6678 . 2 (𝑌 ∈ (har‘𝑋) → 𝑋 ∈ V)
2 reldom 8498 . . . 4 Rel ≼
32brrelex2i 5573 . . 3 (𝑌𝑋𝑋 ∈ V)
43adantl 485 . 2 ((𝑌 ∈ On ∧ 𝑌𝑋) → 𝑋 ∈ V)
5 harval 9008 . . . 4 (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
65eleq2d 2875 . . 3 (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ On ∣ 𝑦𝑋}))
7 breq1 5033 . . . 4 (𝑦 = 𝑌 → (𝑦𝑋𝑌𝑋))
87elrab 3628 . . 3 (𝑌 ∈ {𝑦 ∈ On ∣ 𝑦𝑋} ↔ (𝑌 ∈ On ∧ 𝑌𝑋))
96, 8syl6bb 290 . 2 (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌𝑋)))
101, 4, 9pm5.21nii 383 1 (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2111  {crab 3110  Vcvv 3441   class class class wbr 5030  Oncon0 6159  cfv 6324  cdom 8490  harchar 9004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-wrecs 7930  df-recs 7991  df-en 8493  df-dom 8494  df-oi 8958  df-har 9005
This theorem is referenced by:  harndom  9010  harcard  9391  cardprclem  9392  cardaleph  9500  dfac12lem2  9555  hsmexlem1  9837  pwcfsdom  9994  pwfseqlem5  10074  hargch  10084  harinf  39970  harn0  40041
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