Theorem elharval 9601

Description: The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9599, 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 6944	. 2 ⊢ (𝑌 ∈ (har‘𝑋) → 𝑋 ∈ V)
2		reldom 8991	. . . 4 ⊢ Rel ≼
3	2	brrelex2i 5742	. . 3 ⊢ (𝑌 ≼ 𝑋 → 𝑋 ∈ V)
4	3	adantl 481	. 2 ⊢ ((𝑌 ∈ On ∧ 𝑌 ≼ 𝑋) → 𝑋 ∈ V)
5		harval 9600	. . . 4 ⊢ (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋})
6	5	eleq2d 2827	. . 3 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}))
7		breq1 5146	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ 𝑋 ↔ 𝑌 ≼ 𝑋))
8	7	elrab 3692	. . 3 ⊢ (𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋} ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))
9	6, 8	bitrdi 287	. 2 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)))
10	1, 4, 9	pm5.21nii 378	1 ⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  {crab 3436  Vcvv 3480   class class class wbr 5143  Oncon0 6384  ‘cfv 6561   ≼ cdom 8983  harchar 9596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-en 8986  df-dom 8987  df-oi 9550  df-har 9597

This theorem is referenced by:  harndom  9602  harcard  10018  cardprclem  10019  cardaleph  10129  dfac12lem2  10185  hsmexlem1  10466  pwcfsdom  10623  pwfseqlem5  10703  hargch  10713  harinf  43046  harn0  43114