Theorem elharval 9537

Description: The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9535, 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 ∧ 𝑌 ≼ 𝑋))

Proof of Theorem elharval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6917	. 2 ⊢ (𝑌 ∈ (har‘𝑋) → 𝑋 ∈ V)
2		reldom 8962	. . . 4 ⊢ Rel ≼
3	2	brrelex2i 5716	. . 3 ⊢ (𝑌 ≼ 𝑋 → 𝑋 ∈ V)
4	3	adantl 487	. 2 ⊢ ((𝑌 ∈ On ∧ 𝑌 ≼ 𝑋) → 𝑋 ∈ V)
5		harval 9536	. . . 4 ⊢ (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋})
6	5	eleq2d 2848	. . 3 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}))
7		breq1 5110	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ 𝑋 ↔ 𝑌 ≼ 𝑋))
8	7	elrab 3648	. . 3 ⊢ (𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋} ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))
9	6, 8	bitrdi 290	. 2 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)))
10	1, 4, 9	pm5.21nii 381	1 ⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  {crab 3414  Vcvv 3453   class class class wbr 5107  Oncon0 6361  ‘cfv 6537   ≼ cdom 8954  harchar 9532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-en 8957  df-dom 8958  df-oi 9486  df-har 9533

This theorem is used by:  harndom  9538  harcard  9987  cardprclem  9988  cardaleph  10096  dfac12lem2  10151  hsmexlem1  10432  pwcfsdom  10596  pwfseqlem5  10676  hargch  10686  harinf  43883  harn0  43951