Theorem rankval4 9288

Description: The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.)

Hypothesis

Ref	Expression
rankr1b.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
rankval4	⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankval4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2975	. . . . . 6 ⊢ Ⅎ𝑥𝐴
2		nfcv 2975	. . . . . . 7 ⊢ Ⅎ𝑥𝑅1
3		nfiu1 4944	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
4	2, 3	nffv 6673	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
5	1, 4	dfss2f 3956	. . . . 5 ⊢ (𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
6		vex 3496	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankid 9254	. . . . . 6 ⊢ 𝑥 ∈ (𝑅1‘suc (rank‘𝑥))
8		ssiun2 4962	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
9		rankon 9216	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
10	9	onsuci 7545	. . . . . . . . 9 ⊢ suc (rank‘𝑥) ∈ On
11		rankr1b.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
12	10	rgenw 3148	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
13		iunon 7968	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On)
14	11, 12, 13	mp2an 690	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
15		r1ord3 9203	. . . . . . . . 9 ⊢ ((suc (rank‘𝑥) ∈ On ∧ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On) → (suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
16	10, 14, 15	mp2an 690	. . . . . . . 8 ⊢ (suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
17	8, 16	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
18	17	sseld 3964	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝑅1‘suc (rank‘𝑥)) → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
19	7, 18	mpi 20	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
20	5, 19	mpgbir 1794	. . . 4 ⊢ 𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
21		fvex 6676	. . . . 5 ⊢ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ V
22	21	rankss 9270	. . . 4 ⊢ (𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) → (rank‘𝐴) ⊆ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
23	20, 22	ax-mp 5	. . 3 ⊢ (rank‘𝐴) ⊆ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
24		r1ord3 9203	. . . . . . 7 ⊢ ((∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On ∧ 𝑦 ∈ On) → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
25	14, 24	mpan 688	. . . . . 6 ⊢ (𝑦 ∈ On → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
26	25	ss2rabi 4051	. . . . 5 ⊢ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)}
27		intss 4888	. . . . 5 ⊢ ({𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)} → ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)} ⊆ ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦})
28	26, 27	ax-mp 5	. . . 4 ⊢ ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)} ⊆ ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦}
29		rankval2 9239	. . . . 5 ⊢ ((𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ V → (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) = ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)})
30	21, 29	ax-mp 5	. . . 4 ⊢ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) = ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)}
31		intmin 4887	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On → ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
32	14, 31	ax-mp 5	. . . . 5 ⊢ ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
33	32	eqcomi 2828	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦}
34	28, 30, 33	3sstr4i 4008	. . 3 ⊢ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
35	23, 34	sstri 3974	. 2 ⊢ (rank‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
36		iunss 4960	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
37	11	rankel 9260	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
38		rankon 9216	. . . . 5 ⊢ (rank‘𝐴) ∈ On
39	9, 38	onsucssi 7548	. . . 4 ⊢ ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘𝐴))
40	37, 39	sylib 220	. . 3 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ (rank‘𝐴))
41	36, 40	mprgbir 3151	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴)
42	35, 41	eqssi 3981	1 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)

Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108  ∀wral 3136  {crab 3140  Vcvv 3493   ⊆ wss 3934  ∩ cint 4867  ∪ ciun 4910  Oncon0 6184  suc csuc 6186  ‘cfv 6348  𝑅₁cr1 9183  rankcrnk 9184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-reg 9048  ax-inf2 9096

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-r1 9185  df-rank 9186

This theorem is referenced by:  rankbnd  9289  rankc1  9291  scottrankd  40574