Theorem hartogs 9264

Description: The class of ordinals dominated by a given set is an ordinal. A shorter (when taking into account lemmas hartogslem1 9262 and hartogslem2 9263) proof can be given using the axiom of choice, see ondomon 10303. As its label indicates, this result is used to justify the definition of the Hartogs function df-har 9277. (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
hartogs	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem hartogs

Dummy variables 𝑔 𝑟 𝑠 𝑡 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelon 6288	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
2		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
3		onelss 6305	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
4	3	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ⊆ 𝑧)
5		ssdomg 8757	. . . . . . . . . . . . 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 8764	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≼ 𝐴) → 𝑦 ≼ 𝐴)
9	8	anim2i 616	. . . . . . . . . . . 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 579	. . . . . . . . . 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 5081	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ≼ 𝐴 ↔ 𝑧 ≼ 𝐴))
16	15	elrab 3625	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝐴))
17		breq1 5081	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≼ 𝐴 ↔ 𝑦 ≼ 𝐴))
18	17	elrab 3625	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 ≼ 𝐴))
19	14, 16, 18	3imtr4g 295	. . . . . 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 1802	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
22		dftr2 5197	. . . 4 ⊢ (Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
23	21, 22	mpbir 230	. . 3 ⊢ Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
24		ssrab2 4017	. . 3 ⊢ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ On
25		ordon 7617	. . 3 ⊢ Ord On
26		trssord 6280	. . 3 ⊢ ((Tr {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∧ {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ⊆ On ∧ Ord On) → Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})
27	23, 24, 25, 26	mp3an 1459	. 2 ⊢ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}
28		eqid 2739	. . . 4 ⊢ {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
29		eqid 2739	. . . 4 ⊢ {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑔‘𝑤) ∧ 𝑡 = (𝑔‘𝑧)) ∧ 𝑤 E 𝑧)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑔‘𝑤) ∧ 𝑡 = (𝑔‘𝑧)) ∧ 𝑤 E 𝑧)}
30	28, 29	hartogslem2 9263	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V)
31		elong 6271	. . 3 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V → ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
32	30, 31	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On ↔ Ord {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
33	27, 32	mpbiri 257	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1539   = wceq 1541   ∈ wcel 2109  ∃wrex 3066  {crab 3069  Vcvv 3430   ∖ cdif 3888   ⊆ wss 3891   class class class wbr 5078  {copab 5140  Tr wtr 5195   I cid 5487   E cep 5493   We wwe 5542   × cxp 5586  dom cdm 5588   ↾ cres 5590  Ord word 6262  Oncon0 6263  ‘cfv 6430   ≼ cdom 8705  OrdIsocoi 9229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-en 8708  df-dom 8709  df-oi 9230

This theorem is referenced by:  card2on  9274  harf  9278  harval  9280