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Theorem ondomon 9569
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 8606. (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 5901 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
2 vex 3335 . . . . . . . . . . . . 13 𝑧 ∈ V
3 onelss 5919 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
43imp 444 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
5 ssdomg 8159 . . . . . . . . . . . . 13 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
62, 4, 5mpsyl 68 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
71, 6jca 555 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦𝑧) → (𝑦 ∈ On ∧ 𝑦𝑧))
8 domtr 8166 . . . . . . . . . . . . 13 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
98anim2i 594 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ (𝑦𝑧𝑧𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
109anassrs 683 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
117, 10sylan 489 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
1211exp31 631 . . . . . . . . 9 (𝑧 ∈ On → (𝑦𝑧 → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1312com12 32 . . . . . . . 8 (𝑦𝑧 → (𝑧 ∈ On → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1413impd 446 . . . . . . 7 (𝑦𝑧 → ((𝑧 ∈ On ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴)))
15 breq1 4799 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1615elrab 3496 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑧 ∈ On ∧ 𝑧𝐴))
17 breq1 4799 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817elrab 3496 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
1914, 16, 183imtr4g 285 . . . . . 6 (𝑦𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2019imp 444 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
2120gen2 1864 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
22 dftr2 4898 . . . 4 (Tr {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2321, 22mpbir 221 . . 3 Tr {𝑥 ∈ On ∣ 𝑥𝐴}
24 ssrab2 3820 . . 3 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On
25 ordon 7139 . . 3 Ord On
26 trssord 5893 . . 3 ((Tr {𝑥 ∈ On ∣ 𝑥𝐴} ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥𝐴})
2723, 24, 25, 26mp3an 1565 . 2 Ord {𝑥 ∈ On ∣ 𝑥𝐴}
28 elex 3344 . . . . . 6 (𝐴𝑉𝐴 ∈ V)
29 canth2g 8271 . . . . . . . . 9 (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
30 domsdomtr 8252 . . . . . . . . 9 ((𝑥𝐴𝐴 ≺ 𝒫 𝐴) → 𝑥 ≺ 𝒫 𝐴)
3129, 30sylan2 492 . . . . . . . 8 ((𝑥𝐴𝐴 ∈ V) → 𝑥 ≺ 𝒫 𝐴)
3231expcom 450 . . . . . . 7 (𝐴 ∈ V → (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
3332ralrimivw 3097 . . . . . 6 (𝐴 ∈ V → ∀𝑥 ∈ On (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
3428, 33syl 17 . . . . 5 (𝐴𝑉 → ∀𝑥 ∈ On (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
35 ss2rab 3811 . . . . 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 4991 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
38 numth3 9476 . . . . . 6 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ dom card)
39 cardval2 8999 . . . . . 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 6354 . . . . 5 (card‘𝒫 𝐴) ∈ V
4240, 41syl6eqelr 2840 . . . 4 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V)
43 ssexg 4948 . . . 4 (({𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V) → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
4436, 42, 43syl2anc 696 . . 3 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
45 elong 5884 . . 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 383  wal 1622   = wceq 1624  wcel 2131  wral 3042  {crab 3046  Vcvv 3332  wss 3707  𝒫 cpw 4294   class class class wbr 4796  Tr wtr 4896  dom cdm 5258  Ord word 5875  Oncon0 5876  cfv 6041  cdom 8111  csdm 8112  cardccrd 8943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-ac2 9469
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-wrecs 7568  df-recs 7629  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-card 8947  df-ac 9121
This theorem is referenced by: (None)
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