Theorem rankval3b 9771

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 9740	. . . . . . . . . 10 ⊢ (rank‘𝐴) ∈ On
2		simprl 769	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3		ontri1 6356	. . . . . . . . . 10 ⊢ (((rank‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
4	1, 2, 3	sylancr 587	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
5	4	con2bid 354	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ⊆ 𝑥))
6		r1elssi 9750	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
7	6	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ ∪ (𝑅1 “ On))
8	7	sselda 3947	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ∪ (𝑅1 “ On))
9		rankdmr1 9746	. . . . . . . . . . . . . . . . . 18 ⊢ (rank‘𝐴) ∈ dom 𝑅1
10		r1funlim 9711	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
11	10	simpri 486	. . . . . . . . . . . . . . . . . . 19 ⊢ Lim dom 𝑅1
12		limord 6382	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
13		ordtr1 6365	. . . . . . . . . . . . . . . . . . 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 689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
16	15	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ dom 𝑅1)
17		rankr1ag 9747	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝑦 ∈ (𝑅1‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
18	8, 16, 17	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (𝑅1‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
19	18	ralbidva 3168	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥))
20	19	biimpar 478	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
21	20	an32s 650	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
22		dfss3 3935	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
23	21, 22	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1‘𝑥))
24		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ∈ ∪ (𝑅1 “ On))
25	15	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅1)
26		rankr1bg 9748	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
27	24, 25, 26	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (𝐴 ⊆ (𝑅1‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
28	23, 27	mpbid 231	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (rank‘𝐴) ⊆ 𝑥)
29	28	ex 413	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
30	29	adantrl 714	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
31	5, 30	sylbird 259	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (¬ (rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ⊆ 𝑥))
32	31	pm2.18d 127	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (rank‘𝐴) ⊆ 𝑥)
33	32	ex 413	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
34	33	alrimiv 1930	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35		ssintab 4931	. . . 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 233	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)})
37		df-rab 3406	. . . 4 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
38	37	inteqi 4916	. . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
39	36, 38	sseqtrrdi 3998	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})
40		rankelb 9769	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ 𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
41	40	ralrimiv 3138	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
42		eleq2 2821	. . . . 5 ⊢ (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
43	42	ralbidv 3170	. . . 4 ⊢ (𝑥 = (rank‘𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
44	43	onintss 6373	. . 3 ⊢ ((rank‘𝐴) ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴)))
45	1, 41, 44	mpsyl 68	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴))
46	39, 45	eqssd 3964	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2708  ∀wral 3060  {crab 3405   ⊆ wss 3913  ∪ cuni 4870  ∩ cint 4912  dom cdm 5638   “ cima 5641  Ord word 6321  Oncon0 6322  Lim wlim 6323  Fun wfun 6495  ‘cfv 6501  𝑅₁cr1 9707  rankcrnk 9708

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-r1 9709  df-rank 9710

This theorem is referenced by:  ranksnb  9772  rankonidlem  9773  rankval3  9785  rankunb  9795  rankuni2b  9798  tcrank  9829