Theorem ondomon 10523

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 9504. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.) Use hartogs 9504 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 6360	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
2		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
3		onelss 6377	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
4	3	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
5		ssdomg 8974	. . . . . . . . . . . . 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 8981	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≼ 𝐴) → 𝑦 ≼ 𝐴)
9	8	anim2i 617	. . . . . . . . . . . 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 580	. . . . . . . . . 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 5113	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ≼ 𝐴 ↔ 𝑧 ≼ 𝐴))
16	15	elrab 3662	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝐴))
17		breq1 5113	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐴 ↔ 𝑦 ≼ 𝐴))
18	17	elrab 3662	. . . . . . 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 1796	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
22		dftr2 5219	. . . 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 4046	. . 3 ⊢ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ On
25		ordon 7756	. . 3 ⊢ Ord On
26		trssord 6352	. . 3 ⊢ ((Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
27	23, 24, 25, 26	mp3an 1463	. 2 ⊢ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
28		elex 3471	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
29		canth2g 9101	. . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
30		domsdomtr 9082	. . . . . . . . 9 ⊢ ((𝑥 ≼ 𝐴 ∧ 𝐴 ≺ 𝒫 𝐴) → 𝑥 ≺ 𝒫 𝐴)
31	29, 30	sylan2 593	. . . . . . . 8 ⊢ ((𝑥 ≼ 𝐴 ∧ 𝐴 ∈ V) → 𝑥 ≺ 𝒫 𝐴)
32	31	expcom 413	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
33	32	ralrimivw 3130	. . . . . 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 4037	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ↔ ∀𝑥 ∈ On (𝑥 ≼ 𝐴 → 𝑥 ≺ 𝒫 𝐴))
36	34, 35	sylibr 234	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴})
37		pwexg 5336	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
38		numth3 10430	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ dom card)
39		cardval2 9951	. . . . . 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 6874	. . . . 5 ⊢ (card‘𝒫 𝐴) ∈ V
42	40, 41	eqeltrrdi 2838	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≺ 𝒫 𝐴} ∈ V)
43		ssexg 5281	. . . 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 584	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V)
45		elong 6343	. . 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 258	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ⊆ wss 3917  𝒫 cpw 4566   class class class wbr 5110  Tr wtr 5217  dom cdm 5641  Ord word 6334  Oncon0 6335  ‘cfv 6514   ≼ cdom 8919   ≺ csdm 8920  cardccrd 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-ac2 10423

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-card 9899  df-ac 10076

This theorem is referenced by: (None)