Theorem ondomon 9974

Description: The class of ordinals dominated by a given set is an ordinal. Theorem 56 of [Suppes] p. 227. This theorem can be proved without the axiom of choice, see hartogs 8992. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.) Use hartogs 8992 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ondomon	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ 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.

Step	Hyp	Ref	Expression
1		onelon 6184	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
2		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
3		onelss 6201	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
4	3	imp 410	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
5		ssdomg 8538	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ V → (𝑦 ⊆ 𝑧 → 𝑦 ≼ 𝑧))
6	2, 4, 5	mpsyl 68	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ≼ 𝑧)
7	1, 6	jca 515	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝑧))
8		domtr 8545	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≼ 𝐴) → 𝑦 ≼ 𝐴)
9	8	anim2i 619	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ (𝑦 ≼ 𝑧 ∧ 𝑧 ≼ 𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
10	9	anassrs 471	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≼ 𝑧) ∧ 𝑧 ≼ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
11	7, 10	sylan 583	. . . . . . . . . 10 ⊢ (((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) ∧ 𝑧 ≼ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
12	11	exp31 423	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → (𝑧 ≼ 𝐴 → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))))
13	12	com12 32	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ On → (𝑧 ≼ 𝐴 → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))))
14	13	impd 414	. . . . . . 7 ⊢ (𝑦 ∈ 𝑧 → ((𝑧 ∈ On ∧ 𝑧 ≼ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)))
15		breq1 5033	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ≼ 𝐴 ↔ 𝑧 ≼ 𝐴))
16	15	elrab 3628	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝐴))
17		breq1 5033	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐴 ↔ 𝑦 ≼ 𝐴))
18	17	elrab 3628	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
19	14, 16, 18	3imtr4g 299	. . . . . 6 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
20	19	imp 410	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
21	20	gen2 1798	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
22		dftr2 5138	. . . 4 ⊢ (Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
23	21, 22	mpbir 234	. . 3 ⊢ Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
24		ssrab2 4007	. . 3 ⊢ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ On
25		ordon 7478	. . 3 ⊢ Ord On
26		trssord 6176	. . 3 ⊢ ((Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
27	23, 24, 25, 26	mp3an 1458	. 2 ⊢ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
28		elex 3459	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
29		canth2g 8655	. . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
30		domsdomtr 8636	. . . . . . . . 9 ⊢ ((𝑥 ≼ 𝐴 ∧ 𝐴 ≺ 𝒫 𝐴) → 𝑥 ≺ 𝒫 𝐴)
31	29, 30	sylan2 595	. . . . . . . 8 ⊢ ((𝑥 ≼ 𝐴 ∧ 𝐴 ∈ V) → 𝑥 ≺ 𝒫 𝐴)
32	31	expcom 417	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
33	32	ralrimivw 3150	. . . . . 6 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ On (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
34	28, 33	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ On (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
35		ss2rab 3998	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ↔ ∀𝑥 ∈ On (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
36	34, 35	sylibr 237	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
37		pwexg 5244	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
38		numth3 9881	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ dom card)
39		cardval2 9404	. . . . . 6 ⊢ (𝒫 𝐴 ∈ dom card → (card‘𝒫 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
40	37, 38, 39	3syl 18	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (card‘𝒫 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
41		fvex 6658	. . . . 5 ⊢ (card‘𝒫 𝐴) ∈ V
42	40, 41	eqeltrrdi 2899	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V)
43		ssexg 5191	. . . 4 ⊢ (({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V) → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V)
44	36, 42, 43	syl2anc 587	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V)
45		elong 6167	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V → ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
46	44, 45	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
47	27, 46	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497   class class class wbr 5030  Tr wtr 5136  dom cdm 5519  Ord word 6158  Oncon0 6159  ‘cfv 6324   ≼ cdom 8490   ≺ csdm 8491  cardccrd 9348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-ac2 9874

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-wrecs 7930  df-recs 7991  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-card 9352  df-ac 9527

This theorem is referenced by: (None)