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Theorem elharval 8635
 Description: The Hartogs number of a set is greater than all ordinals which inject into it. (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 6383 . 2 (𝑌 ∈ (har‘𝑋) → 𝑋 ∈ V)
2 reldom 8129 . . . 4 Rel ≼
32brrelex2i 5316 . . 3 (𝑌𝑋𝑋 ∈ V)
43adantl 473 . 2 ((𝑌 ∈ On ∧ 𝑌𝑋) → 𝑋 ∈ V)
5 harval 8634 . . . 4 (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
65eleq2d 2825 . . 3 (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ On ∣ 𝑦𝑋}))
7 breq1 4807 . . . 4 (𝑦 = 𝑌 → (𝑦𝑋𝑌𝑋))
87elrab 3504 . . 3 (𝑌 ∈ {𝑦 ∈ On ∣ 𝑦𝑋} ↔ (𝑌 ∈ On ∧ 𝑌𝑋))
96, 8syl6bb 276 . 2 (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌𝑋)))
101, 4, 9pm5.21nii 367 1 (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌𝑋))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∈ wcel 2139  {crab 3054  Vcvv 3340   class class class wbr 4804  Oncon0 5884  ‘cfv 6049   ≼ cdom 8121  harchar 8628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-wrecs 7577  df-recs 7638  df-en 8124  df-dom 8125  df-oi 8582  df-har 8630 This theorem is referenced by:  harndom  8636  harcard  9014  cardprclem  9015  cardaleph  9122  dfac12lem2  9178  hsmexlem1  9460  pwcfsdom  9617  pwfseqlem5  9697  hargch  9707  harinf  38121  harn0  38192
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