Theorem rankr1c 9579

Description: A relationship between the rank function and the cumulative hierarchy of sets function 𝑅₁. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
rankr1c	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))

Proof of Theorem rankr1c

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝐵 = (rank‘𝐴) → 𝐵 = (rank‘𝐴))
2		rankdmr1 9559	. . . 4 ⊢ (rank‘𝐴) ∈ dom 𝑅1
3	1, 2	eqeltrdi 2847	. . 3 ⊢ (𝐵 = (rank‘𝐴) → 𝐵 ∈ dom 𝑅1)
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 = (rank‘𝐴) → 𝐵 ∈ dom 𝑅1))
5		elfvdm 6806	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘suc 𝐵) → suc 𝐵 ∈ dom 𝑅1)
6		r1funlim 9524	. . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
7	6	simpri 486	. . . . . 6 ⊢ Lim dom 𝑅1
8		limsuc 7696	. . . . . 6 ⊢ (Lim dom 𝑅1 → (𝐵 ∈ dom 𝑅1 ↔ suc 𝐵 ∈ dom 𝑅1))
9	7, 8	ax-mp 5	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 ↔ suc 𝐵 ∈ dom 𝑅1)
10	5, 9	sylibr 233	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐵 ∈ dom 𝑅1)
11	10	adantl 482	. . 3 ⊢ ((¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → 𝐵 ∈ dom 𝑅1)
12	11	a1i 11	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → 𝐵 ∈ dom 𝑅1))
13		eqss 3936	. . . 4 ⊢ (𝐵 = (rank‘𝐴) ↔ (𝐵 ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ⊆ 𝐵))
14		rankr1clem 9578	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
15		rankr1ag 9560	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵))
16	9, 15	sylan2b 594	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵))
17		rankon 9553	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
18		limord 6325	. . . . . . . . . 10 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
19	7, 18	ax-mp 5	. . . . . . . . 9 ⊢ Ord dom 𝑅1
20		ordelon 6290	. . . . . . . . 9 ⊢ ((Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On)
21	19, 20	mpan 687	. . . . . . . 8 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On)
22	21	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On)
23		onsssuc 6353	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵))
24	17, 22, 23	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵))
25	16, 24	bitr4d 281	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))
26	14, 25	anbi12d 631	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) ↔ (𝐵 ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ⊆ 𝐵)))
27	13, 26	bitr4id 290	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))
28	27	ex 413	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 ∈ dom 𝑅1 → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)))))
29	4, 12, 28	pm5.21ndd 381	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ∪ cuni 4839  dom cdm 5589   “ cima 5592  Ord word 6265  Oncon0 6266  Lim wlim 6267  suc csuc 6268  Fun wfun 6427  ‘cfv 6433  𝑅₁cr1 9520  rankcrnk 9521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523

This theorem is referenced by:  rankidn  9580  rankpwi  9581  rankr1g  9590  r1tskina  10538