Theorem rankelb 9720

Description: The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
rankelb	⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))

Proof of Theorem rankelb

Step	Hyp	Ref	Expression
1		r1elssi 9701	. . . . . 6 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ ∪ (𝑅1 “ On))
2	1	sseld 3934	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ (𝑅1 “ On)))
3		rankidn 9718	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
4	2, 3	syl6 35	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
5	4	imp 406	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
6		rankon 9691	. . . . 5 ⊢ (rank‘𝐵) ∈ On
7		rankon 9691	. . . . 5 ⊢ (rank‘𝐴) ∈ On
8		ontri1 6341	. . . . 5 ⊢ (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)))
9	6, 7, 8	mp2an 692	. . . 4 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))
10		rankdmr1 9697	. . . . . 6 ⊢ (rank‘𝐵) ∈ dom 𝑅1
11		rankdmr1 9697	. . . . . 6 ⊢ (rank‘𝐴) ∈ dom 𝑅1
12		r1ord3g 9675	. . . . . 6 ⊢ (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))))
13	10, 11, 12	mp2an 692	. . . . 5 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))
14		r1rankidb 9700	. . . . . 6 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
15	14	sselda 3935	. . . . 5 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐵)))
16		ssel 3929	. . . . 5 ⊢ ((𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)) → (𝐴 ∈ (𝑅1‘(rank‘𝐵)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
17	13, 15, 16	syl2imc 41	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → ((rank‘𝐵) ⊆ (rank‘𝐴) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
18	9, 17	biimtrrid 243	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → (¬ (rank‘𝐴) ∈ (rank‘𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
19	5, 18	mt3d 148	. 2 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
20	19	ex 412	1 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ⊆ wss 3903  ∪ cuni 4858  dom cdm 5619   “ cima 5622  Oncon0 6307  ‘cfv 6482  𝑅₁cr1 9658  rankcrnk 9659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-r1 9660  df-rank 9661

This theorem is referenced by:  wfelirr  9721  rankval3b  9722  rankel  9735  rankunb  9746  rankuni2b  9749  rankcf  10671  rankrelp  44954