Theorem elharval 9410

Description: The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9408, 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 6857	. 2 ⊢ (𝑌 ∈ (har‘𝑋) → 𝑋 ∈ V)
2		reldom 8802	. . . 4 ⊢ Rel ≼
3	2	brrelex2i 5669	. . 3 ⊢ (𝑌 ≼ 𝑋 → 𝑋 ∈ V)
4	3	adantl 482	. 2 ⊢ ((𝑌 ∈ On ∧ 𝑌 ≼ 𝑋) → 𝑋 ∈ V)
5		harval 9409	. . . 4 ⊢ (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋})
6	5	eleq2d 2822	. . 3 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}))
7		breq1 5092	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ 𝑋 ↔ 𝑌 ≼ 𝑋))
8	7	elrab 3634	. . 3 ⊢ (𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋} ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))
9	6, 8	bitrdi 286	. 2 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)))
10	1, 4, 9	pm5.21nii 379	1 ⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2105  {crab 3403  Vcvv 3441   class class class wbr 5089  Oncon0 6296  ‘cfv 6473   ≼ cdom 8794  harchar 9405

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 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

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 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-en 8797  df-dom 8798  df-oi 9359  df-har 9406

This theorem is referenced by:  harndom  9411  harcard  9827  cardprclem  9828  cardaleph  9938  dfac12lem2  9993  hsmexlem1  10275  pwcfsdom  10432  pwfseqlem5  10512  hargch  10522  harinf  41107  harn0  41178