rankval3b - Metamath Proof Explorer

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Theorem rankval3b 9825

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 9794	. . . . . . . . . 10 ⊢ (rank‘𝐴) ∈ On
2		simprl 768	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3		ontri1 6398	. . . . . . . . . 10 ⊢ (((rank‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
4	1, 2, 3	sylancr 586	. . . . . . . . 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 9804	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → 𝐴 ⊆ ∪ (𝑅₁ “ On))
7	6	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ ∪ (𝑅₁ “ On))
8	7	sselda 3982	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ∪ (𝑅₁ “ On))
9		rankdmr1 9800	. . . . . . . . . . . . . . . . . 18 ⊢ (rank‘𝐴) ∈ dom 𝑅₁
10		r1funlim 9765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun 𝑅₁ ∧ Lim dom 𝑅₁)
11	10	simpri 485	. . . . . . . . . . . . . . . . . . 19 ⊢ Lim dom 𝑅₁
12		limord 6424	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim dom 𝑅₁ → Ord dom 𝑅₁)
13		ordtr1 6407	. . . . . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅₁)
16	15	ad2antlr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ dom 𝑅₁)
17		rankr1ag 9801	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ∪ (𝑅₁ “ On) ∧ 𝑥 ∈ dom 𝑅₁) → (𝑦 ∈ (𝑅₁‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
18	8, 16, 17	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (𝑅₁‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
19	18	ralbidva 3174	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅₁‘𝑥) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥))
20	19	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅₁‘𝑥))
21	20	an32s 649	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅₁‘𝑥))
22		dfss3 3970	. . . . . . . . . . . 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 764	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ∈ ∪ (𝑅₁ “ On))
25	15	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅₁)
26		rankr1bg 9802	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ 𝑥 ∈ dom 𝑅₁) → (𝐴 ⊆ (𝑅₁‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
27	24, 25, 26	syl2anc 583	. . . . . . . . . . 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 412	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
30	29	adantrl 713	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
31	5, 30	sylbird 260	. . . . . . 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 412	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
34	33	alrimiv 1929	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35		ssintab 4969	. . . 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 3432	. . . 4 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
38	37	inteqi 4954	. . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
39	36, 38	sseqtrrdi 4033	. 2 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})
40		rankelb 9823	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (𝑦 ∈ 𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
41	40	ralrimiv 3144	. . 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 3176	. . . 4 ⊢ (𝑥 = (rank‘𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
44	43	onintss 6415	. . 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 3999	1 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2105 {cab 2708 ∀wral 3060 {crab 3431 ⊆ wss 3948 ∪ cuni 4908 ∩ cint 4950 dom cdm 5676 “ cima 5679 Ord word 6363 Oncon0 6364 Lim wlim 6365 Fun wfun 6537 ‘cfv 6543 𝑅₁cr1 9761 rankcrnk 9762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-r1 9763 df-rank 9764

This theorem is referenced by: ranksnb 9826 rankonidlem 9827 rankval3 9839 rankunb 9849 rankuni2b 9852 tcrank 9883

W3C validator