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Theorem hartogs 8393
Description: Given any set, the Hartogs number of the set is the least ordinal not dominated by that set. This theorem proves that there is always an ordinal which satisfies this. (This theorem can be proven trivially using the AC - see theorem ondomon 9329- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
hartogs (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem hartogs
Dummy variables 𝑔 𝑟 𝑠 𝑡 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 5707 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
2 vex 3189 . . . . . . . . . . . . 13 𝑧 ∈ V
3 onelss 5725 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
43imp 445 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
5 ssdomg 7945 . . . . . . . . . . . . 13 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
62, 4, 5mpsyl 68 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
71, 6jca 554 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦𝑧) → (𝑦 ∈ On ∧ 𝑦𝑧))
8 domtr 7953 . . . . . . . . . . . . 13 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
98anim2i 592 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ (𝑦𝑧𝑧𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
109anassrs 679 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
117, 10sylan 488 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
1211exp31 629 . . . . . . . . 9 (𝑧 ∈ On → (𝑦𝑧 → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1312com12 32 . . . . . . . 8 (𝑦𝑧 → (𝑧 ∈ On → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1413impd 447 . . . . . . 7 (𝑦𝑧 → ((𝑧 ∈ On ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴)))
15 breq1 4616 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1615elrab 3346 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑧 ∈ On ∧ 𝑧𝐴))
17 breq1 4616 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817elrab 3346 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
1914, 16, 183imtr4g 285 . . . . . 6 (𝑦𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2019imp 445 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
2120gen2 1720 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
22 dftr2 4714 . . . 4 (Tr {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2321, 22mpbir 221 . . 3 Tr {𝑥 ∈ On ∣ 𝑥𝐴}
24 ssrab2 3666 . . 3 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On
25 ordon 6929 . . 3 Ord On
26 trssord 5699 . . 3 ((Tr {𝑥 ∈ On ∣ 𝑥𝐴} ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥𝐴})
2723, 24, 25, 26mp3an 1421 . 2 Ord {𝑥 ∈ On ∣ 𝑥𝐴}
28 eqid 2621 . . . 4 {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
29 eqid 2621 . . . 4 {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑔𝑤) ∧ 𝑡 = (𝑔𝑧)) ∧ 𝑤 E 𝑧)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑔𝑤) ∧ 𝑡 = (𝑔𝑧)) ∧ 𝑤 E 𝑧)}
3028, 29hartogslem2 8392 . . 3 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
31 elong 5690 . . 3 ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ V → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥𝐴}))
3230, 31syl 17 . 2 (𝐴𝑉 → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥𝐴}))
3327, 32mpbiri 248 1 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wrex 2908  {crab 2911  Vcvv 3186  cdif 3552  wss 3555   class class class wbr 4613  {copab 4672  Tr wtr 4712   E cep 4983   I cid 4984   We wwe 5032   × cxp 5072  dom cdm 5074  cres 5076  Ord word 5681  Oncon0 5682  cfv 5847  cdom 7897  OrdIsocoi 8358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-wrecs 7352  df-recs 7413  df-en 7900  df-dom 7901  df-oi 8359
This theorem is referenced by:  card2on  8403  harf  8409  harval  8411
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