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Theorem ondomon 9329
Description: The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227. This theorem can be proved (with a longer proof) without the Axiom of Choice; see hartogs 8393. (Contributed by NM, 7-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ondomon (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem ondomon
Dummy variables 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 elex 3198 . . . . . 6 (𝐴𝑉𝐴 ∈ V)
29 canth2g 8058 . . . . . . . . 9 (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
30 domsdomtr 8039 . . . . . . . . 9 ((𝑥𝐴𝐴 ≺ 𝒫 𝐴) → 𝑥 ≺ 𝒫 𝐴)
3129, 30sylan2 491 . . . . . . . 8 ((𝑥𝐴𝐴 ∈ V) → 𝑥 ≺ 𝒫 𝐴)
3231expcom 451 . . . . . . 7 (𝐴 ∈ V → (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
3332ralrimivw 2961 . . . . . 6 (𝐴 ∈ V → ∀𝑥 ∈ On (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
3428, 33syl 17 . . . . 5 (𝐴𝑉 → ∀𝑥 ∈ On (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
35 ss2rab 3657 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ↔ ∀𝑥 ∈ On (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
3634, 35sylibr 224 . . . 4 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
37 pwexg 4810 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
38 numth3 9236 . . . . . 6 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ dom card)
39 cardval2 8761 . . . . . 6 (𝒫 𝐴 ∈ dom card → (card‘𝒫 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
4037, 38, 393syl 18 . . . . 5 (𝐴𝑉 → (card‘𝒫 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
41 fvex 6158 . . . . 5 (card‘𝒫 𝐴) ∈ V
4240, 41syl6eqelr 2707 . . . 4 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V)
43 ssexg 4764 . . . 4 (({𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V) → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
4436, 42, 43syl2anc 692 . . 3 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
45 elong 5690 . . 3 ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ V → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥𝐴}))
4644, 45syl 17 . 2 (𝐴𝑉 → ({𝑥 ∈ On ∣ 𝑥𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥𝐴}))
4727, 46mpbiri 248 1 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   class class class wbr 4613  Tr wtr 4712  dom cdm 5074  Ord word 5681  Oncon0 5682  cfv 5847  cdom 7897  csdm 7898  cardccrd 8705
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  ax-ac2 9229
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-int 4441  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-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-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-card 8709  df-ac 8883
This theorem is referenced by: (None)
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