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Theorem elharval 9418
Description: The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9416, 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 (π‘Œ ∈ (harβ€˜π‘‹) ↔ (π‘Œ ∈ On ∧ π‘Œ β‰Ό 𝑋))

Proof of Theorem elharval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elfvex 6863 . 2 (π‘Œ ∈ (harβ€˜π‘‹) β†’ 𝑋 ∈ V)
2 reldom 8810 . . . 4 Rel β‰Ό
32brrelex2i 5675 . . 3 (π‘Œ β‰Ό 𝑋 β†’ 𝑋 ∈ V)
43adantl 482 . 2 ((π‘Œ ∈ On ∧ π‘Œ β‰Ό 𝑋) β†’ 𝑋 ∈ V)
5 harval 9417 . . . 4 (𝑋 ∈ V β†’ (harβ€˜π‘‹) = {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝑋})
65eleq2d 2822 . . 3 (𝑋 ∈ V β†’ (π‘Œ ∈ (harβ€˜π‘‹) ↔ π‘Œ ∈ {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝑋}))
7 breq1 5095 . . . 4 (𝑦 = π‘Œ β†’ (𝑦 β‰Ό 𝑋 ↔ π‘Œ β‰Ό 𝑋))
87elrab 3634 . . 3 (π‘Œ ∈ {𝑦 ∈ On ∣ 𝑦 β‰Ό 𝑋} ↔ (π‘Œ ∈ On ∧ π‘Œ β‰Ό 𝑋))
96, 8bitrdi 286 . 2 (𝑋 ∈ V β†’ (π‘Œ ∈ (harβ€˜π‘‹) ↔ (π‘Œ ∈ On ∧ π‘Œ β‰Ό 𝑋)))
101, 4, 9pm5.21nii 379 1 (π‘Œ ∈ (harβ€˜π‘‹) ↔ (π‘Œ ∈ On ∧ π‘Œ β‰Ό 𝑋))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2105  {crab 3403  Vcvv 3441   class class class wbr 5092  Oncon0 6302  β€˜cfv 6479   β‰Ό cdom 8802  harchar 9413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-en 8805  df-dom 8806  df-oi 9367  df-har 9414
This theorem is referenced by:  harndom  9419  harcard  9835  cardprclem  9836  cardaleph  9946  dfac12lem2  10001  hsmexlem1  10283  pwcfsdom  10440  pwfseqlem5  10520  hargch  10530  harinf  41119  harn0  41190
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