Theorem rankval3b 9863

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 9830	. . . . . . . . . 10 ⊢ (rank‘𝐴) ∈ On
2		simprl 783	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3		ontri1 6415	. . . . . . . . . 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 9840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
7	6	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ ∪ (𝑅1 “ On))
8	7	sselda 3913	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ∪ (𝑅1 “ On))
9		rankdmr1 9836	. . . . . . . . . . . . . . . . . 18 ⊢ (rank‘𝐴) ∈ dom 𝑅1
10		r1dmlim 9799	. . . . . . . . . . . . . . . . . . 19 ⊢ Lim dom 𝑅1
11		limord 6442	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
12		ordtr1 6425	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord dom 𝑅1 → ((𝑥 ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
13	10, 11, 12	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
14	9, 13	mpan2 704	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
15	14	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ dom 𝑅1)
16		rankr1ag 9837	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝑦 ∈ (𝑅1‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
17	8, 15, 16	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (𝑅1‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
18	17	ralbidva 3187	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥))
19	18	biimpar 483	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
20	19	an32s 665	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
21		dfss3 3902	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
22	20, 21	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1‘𝑥))
23		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ∈ ∪ (𝑅1 “ On))
24	14	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅1)
25		rankr1bg 9838	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
26	23, 24, 25	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (𝐴 ⊆ (𝑅1‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
27	22, 26	mpbid 235	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (rank‘𝐴) ⊆ 𝑥)
28	27	ex 418	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
29	28	adantrl 729	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
30	5, 29	sylbird 263	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (¬ (rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ⊆ 𝑥))
31	30	pm2.18d 128	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (rank‘𝐴) ⊆ 𝑥)
32	31	ex 418	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
33	32	alrimiv 1962	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
34		ssintab 4928	. . . 4 ⊢ ((rank‘𝐴) ⊆ ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35	33, 34	sylibr 237	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)})
36		df-rab 3417	. . . 4 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
37	36	inteqi 4914	. . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
38	35, 37	sseqtrrdi 3954	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})
39		rankelb 9860	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ 𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
40	39	ralrimiv 3157	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
41		eleq2 2853	. . . . 5 ⊢ (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
42	41	ralbidv 3189	. . . 4 ⊢ (𝑥 = (rank‘𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
43	42	onintss 6433	. . 3 ⊢ ((rank‘𝐴) ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴)))
44	1, 40, 43	mpsyl 69	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴))
45	38, 44	eqssd 3930	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 2147  {cab 2741  ∀wral 3080  {crab 3416   ⊆ wss 3900  ∪ cuni 4870  ∩ cint 4910  dom cdm 5655   “ cima 5658  Ord word 6379  Oncon0 6380  Lim wlim 6381  ‘cfv 6558  𝑅₁cr1 9793  rankcrnk 9794

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-r1 9795  df-rank 9796

This theorem is used by:  ranksnb  9864  rankonidlem  9865  rankval3  9880  rankunb  9891  rankuni2b  9894  tcrank  9928