Theorem rankelb 9750

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 9731	. . . . . 6 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ ∪ (𝑅1 “ On))
2	1	sseld 3934	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ (𝑅1 “ On)))
3		rankidn 9748	. . . . 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 9721	. . . . 5 ⊢ (rank‘𝐵) ∈ On
7		rankon 9721	. . . . 5 ⊢ (rank‘𝐴) ∈ On
8		ontri1 6361	. . . . 5 ⊢ (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)))
9	6, 7, 8	mp2an 693	. . . 4 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))
10		rankdmr1 9727	. . . . . 6 ⊢ (rank‘𝐵) ∈ dom 𝑅1
11		rankdmr1 9727	. . . . . 6 ⊢ (rank‘𝐴) ∈ dom 𝑅1
12		r1ord3g 9705	. . . . . 6 ⊢ (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))))
13	10, 11, 12	mp2an 693	. . . . 5 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))
14		r1rankidb 9730	. . . . . 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 2114   ⊆ wss 3903  ∪ cuni 4865  dom cdm 5634   “ cima 5637  Oncon0 6327  ‘cfv 6502  𝑅₁cr1 9688  rankcrnk 9689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-r1 9690  df-rank 9691

This theorem is referenced by:  wfelirr  9751  rankval3b  9752  rankel  9765  rankunb  9776  rankuni2b  9779  rankcf  10702  r1elcl  35281  rankval4b  35283  rankfilimb  35285  regsfromunir1  36698  rankrelp  45345