Theorem ondomon 10477

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 9453. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.) Use hartogs 9453 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 6343	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
2		vex 3445	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
3		onelss 6360	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
4	3	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
5		ssdomg 8941	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ V → (𝑦 ⊆ 𝑧 → 𝑦 ≼ 𝑧))
6	2, 4, 5	mpsyl 68	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ≼ 𝑧)
7	1, 6	jca 511	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝑧))
8		domtr 8948	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≼ 𝐴) → 𝑦 ≼ 𝐴)
9	8	anim2i 618	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ (𝑦 ≼ 𝑧 ∧ 𝑧 ≼ 𝐴)) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
10	9	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≼ 𝑧) ∧ 𝑧 ≼ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
11	7, 10	sylan 581	. . . . . . . . . 10 ⊢ (((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) ∧ 𝑧 ≼ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
12	11	exp31 419	. . . . . . . . 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 410	. . . . . . 7 ⊢ (𝑦 ∈ 𝑧 → ((𝑧 ∈ On ∧ 𝑧 ≼ 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴)))
15		breq1 5102	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ≼ 𝐴 ↔ 𝑧 ≼ 𝐴))
16	15	elrab 3647	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝐴))
17		breq1 5102	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐴 ↔ 𝑦 ≼ 𝐴))
18	17	elrab 3647	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
19	14, 16, 18	3imtr4g 296	. . . . . 6 ⊢ (𝑦 ∈ 𝑧 → (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
20	19	imp 406	. . . . 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 5208	. . . 4 ⊢ (Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
23	21, 22	mpbir 231	. . 3 ⊢ Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
24		ssrab2 4033	. . 3 ⊢ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ On
25		ordon 7724	. . 3 ⊢ Ord On
26		trssord 6335	. . 3 ⊢ ((Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
27	23, 24, 25, 26	mp3an 1464	. 2 ⊢ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
28		pwexg 5324	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
29		numth3 10384	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ dom card)
30		cardval2 9907	. . . . . 6 ⊢ (𝒫 𝐴 ∈ dom card → (card‘𝒫 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (card‘𝒫 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
32		fvex 6848	. . . . 5 ⊢ (card‘𝒫 𝐴) ∈ V
33	31, 32	eqeltrrdi 2846	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V)
34		elex 3462	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
35		canth2g 9063	. . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
36		domsdomtr 9044	. . . . . . . . 9 ⊢ ((𝑥 ≼ 𝐴 ∧ 𝐴 ≺ 𝒫 𝐴) → 𝑥 ≺ 𝒫 𝐴)
37	35, 36	sylan2 594	. . . . . . . 8 ⊢ ((𝑥 ≼ 𝐴 ∧ 𝐴 ∈ V) → 𝑥 ≺ 𝒫 𝐴)
38	37	expcom 413	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
39	38	ralrimivw 3133	. . . . . 6 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ On (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
40	34, 39	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ On (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
41	40	ss2rabd 4025	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
42	33, 41	ssexd 5270	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V)
43		elong 6326	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V → ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
44	42, 43	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
45	27, 44	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400  Vcvv 3441   ⊆ wss 3902  𝒫 cpw 4555   class class class wbr 5099  Tr wtr 5206  dom cdm 5625  Ord word 6317  Oncon0 6318  ‘cfv 6493   ≼ cdom 8885   ≺ csdm 8886  cardccrd 9851

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-ac2 10377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-card 9855  df-ac 10030

This theorem is referenced by: (None)