MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ondomon Structured version   Visualization version   GIF version

Theorem ondomon 9974
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 8997. (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 6214 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
2 vex 3503 . . . . . . . . . . . . 13 𝑧 ∈ V
3 onelss 6231 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
43imp 407 . . . . . . . . . . . . 13 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
5 ssdomg 8544 . . . . . . . . . . . . 13 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
62, 4, 5mpsyl 68 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦𝑧)
71, 6jca 512 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦𝑧) → (𝑦 ∈ On ∧ 𝑦𝑧))
8 domtr 8551 . . . . . . . . . . . . 13 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
98anim2i 616 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ (𝑦𝑧𝑧𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
109anassrs 468 . . . . . . . . . . 11 (((𝑦 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
117, 10sylan 580 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑦𝑧) ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴))
1211exp31 420 . . . . . . . . 9 (𝑧 ∈ On → (𝑦𝑧 → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1312com12 32 . . . . . . . 8 (𝑦𝑧 → (𝑧 ∈ On → (𝑧𝐴 → (𝑦 ∈ On ∧ 𝑦𝐴))))
1413impd 411 . . . . . . 7 (𝑦𝑧 → ((𝑧 ∈ On ∧ 𝑧𝐴) → (𝑦 ∈ On ∧ 𝑦𝐴)))
15 breq1 5066 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1615elrab 3684 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑧 ∈ On ∧ 𝑧𝐴))
17 breq1 5066 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817elrab 3684 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
1914, 16, 183imtr4g 297 . . . . . 6 (𝑦𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2019imp 407 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
2120gen2 1790 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴})
22 dftr2 5171 . . . 4 (Tr {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴}))
2321, 22mpbir 232 . . 3 Tr {𝑥 ∈ On ∣ 𝑥𝐴}
24 ssrab2 4060 . . 3 {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On
25 ordon 7486 . . 3 Ord On
26 trssord 6206 . . 3 ((Tr {𝑥 ∈ On ∣ 𝑥𝐴} ∧ {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥𝐴})
2723, 24, 25, 26mp3an 1454 . 2 Ord {𝑥 ∈ On ∣ 𝑥𝐴}
28 elex 3518 . . . . . 6 (𝐴𝑉𝐴 ∈ V)
29 canth2g 8660 . . . . . . . . 9 (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
30 domsdomtr 8641 . . . . . . . . 9 ((𝑥𝐴𝐴 ≺ 𝒫 𝐴) → 𝑥 ≺ 𝒫 𝐴)
3129, 30sylan2 592 . . . . . . . 8 ((𝑥𝐴𝐴 ∈ V) → 𝑥 ≺ 𝒫 𝐴)
3231expcom 414 . . . . . . 7 (𝐴 ∈ V → (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
3332ralrimivw 3188 . . . . . 6 (𝐴 ∈ V → ∀𝑥 ∈ On (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
3428, 33syl 17 . . . . 5 (𝐴𝑉 → ∀𝑥 ∈ On (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
35 ss2rab 4051 . . . . 5 ({𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ↔ ∀𝑥 ∈ On (𝑥𝐴𝑥 ≺ 𝒫 𝐴))
3634, 35sylibr 235 . . . 4 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
37 pwexg 5276 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
38 numth3 9881 . . . . . 6 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ dom card)
39 cardval2 9409 . . . . . 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 6680 . . . . 5 (card‘𝒫 𝐴) ∈ V
4240, 41syl6eqelr 2927 . . . 4 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V)
43 ssexg 5224 . . . 4 (({𝑥 ∈ On ∣ 𝑥𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V) → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
4436, 42, 43syl2anc 584 . . 3 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)
45 elong 6197 . . 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 259 1 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  wss 3940  𝒫 cpw 4542   class class class wbr 5063  Tr wtr 5169  dom cdm 5554  Ord word 6188  Oncon0 6189  cfv 6352  cdom 8496  csdm 8497  cardccrd 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-ac2 9874
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-wrecs 7938  df-recs 7999  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-card 9357  df-ac 9531
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator