Theorem rankr1c 9714

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 9694	. . . 4 ⊢ (rank‘𝐴) ∈ dom 𝑅1
3	1, 2	eqeltrdi 2839	. . 3 ⊢ (𝐵 = (rank‘𝐴) → 𝐵 ∈ dom 𝑅1)
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 = (rank‘𝐴) → 𝐵 ∈ dom 𝑅1))
5		elfvdm 6856	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘suc 𝐵) → suc 𝐵 ∈ dom 𝑅1)
6		r1funlim 9659	. . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
7	6	simpri 485	. . . . . 6 ⊢ Lim dom 𝑅1
8		limsuc 7779	. . . . . 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 234	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐵 ∈ dom 𝑅1)
11	10	adantl 481	. . 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 3945	. . . 4 ⊢ (𝐵 = (rank‘𝐴) ↔ (𝐵 ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ⊆ 𝐵))
14		rankr1clem 9713	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))
15		rankr1ag 9695	. . . . . . 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 9688	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
18		limord 6367	. . . . . . . . . 10 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
19	7, 18	ax-mp 5	. . . . . . . . 9 ⊢ Ord dom 𝑅1
20		ordelon 6330	. . . . . . . . 9 ⊢ ((Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On)
21	19, 20	mpan 690	. . . . . . . 8 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On)
22	21	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On)
23		onsssuc 6398	. . . . . . 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 282	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))
26	14, 25	anbi12d 632	. . . 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 412	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 ∈ dom 𝑅1 → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)))))
29	4, 12, 28	pm5.21ndd 379	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∪ cuni 4856  dom cdm 5614   “ cima 5617  Ord word 6305  Oncon0 6306  Lim wlim 6307  suc csuc 6308  Fun wfun 6475  ‘cfv 6481  𝑅₁cr1 9655  rankcrnk 9656

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-r1 9657  df-rank 9658

This theorem is referenced by:  rankidn  9715  rankpwi  9716  rankr1g  9725  r1tskina  10673