Theorem rankval3b 9809

Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
rankval3b	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem rankval3b

Step	Hyp	Ref	Expression
1		rankon 9777	. . . . . . . . . 10 ⊢ (rank‘𝐴) ∈ On
2		simprl 783	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3		ontri1 6386	. . . . . . . . . 10 ⊢ (((rank‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
4	1, 2, 3	sylancr 599	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
5	4	con2bid 357	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ⊆ 𝑥))
6		r1elssi 9787	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
7	6	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ ∪ (𝑅1 “ On))
8	7	sselda 3930	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ∪ (𝑅1 “ On))
9		rankdmr1 9783	. . . . . . . . . . . . . . . . . 18 ⊢ (rank‘𝐴) ∈ dom 𝑅1
10		r1funlim 9748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
11	10	simpri 491	. . . . . . . . . . . . . . . . . . 19 ⊢ Lim dom 𝑅1
12		limord 6413	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
13		ordtr1 6396	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord dom 𝑅1 → ((𝑥 ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
14	11, 12, 13	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
15	9, 14	mpan2 704	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
16	15	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ dom 𝑅1)
17		rankr1ag 9784	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝑦 ∈ (𝑅1‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
18	8, 16, 17	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (𝑅1‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
19	18	ralbidva 3183	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥))
20	19	biimpar 483	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
21	20	an32s 665	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
22		dfss3 3919	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
23	21, 22	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1‘𝑥))
24		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ∈ ∪ (𝑅1 “ On))
25	15	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅1)
26		rankr1bg 9785	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
27	24, 25, 26	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (𝐴 ⊆ (𝑅1‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
28	23, 27	mpbid 235	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (rank‘𝐴) ⊆ 𝑥)
29	28	ex 418	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
30	29	adantrl 729	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
31	5, 30	sylbird 263	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (¬ (rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ⊆ 𝑥))
32	31	pm2.18d 128	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (rank‘𝐴) ⊆ 𝑥)
33	32	ex 418	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
34	33	alrimiv 1960	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35		ssintab 4924	. . . 4 ⊢ ((rank‘𝐴) ⊆ ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
36	34, 35	sylibr 237	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)})
37		df-rab 3413	. . . 4 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
38	37	inteqi 4910	. . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
39	36, 38	sseqtrrdi 3971	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})
40		rankelb 9806	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ 𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
41	40	ralrimiv 3153	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
42		eleq2 2849	. . . . 5 ⊢ (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
43	42	ralbidv 3185	. . . 4 ⊢ (𝑥 = (rank‘𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
44	43	onintss 6404	. . 3 ⊢ ((rank‘𝐴) ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴)))
45	1, 41, 44	mpsyl 69	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴))
46	39, 45	eqssd 3947	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  {crab 3412   ⊆ wss 3898  ∪ cuni 4866  ∩ cint 4906  dom cdm 5647   “ cima 5650  Ord word 6350  Oncon0 6351  Lim wlim 6352  Fun wfun 6521  ‘cfv 6527  𝑅₁cr1 9744  rankcrnk 9745

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9746  df-rank 9747

This theorem is used by:  ranksnb  9810  rankonidlem  9811  rankval3  9826  rankunb  9837  rankuni2b  9840  tcrank  9874