Theorem rankelb 9582

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 9563	. . . . . 6 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ ∪ (𝑅1 “ On))
2	1	sseld 3920	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ (𝑅1 “ On)))
3		rankidn 9580	. . . . 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 407	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
6		rankon 9553	. . . . 5 ⊢ (rank‘𝐵) ∈ On
7		rankon 9553	. . . . 5 ⊢ (rank‘𝐴) ∈ On
8		ontri1 6300	. . . . 5 ⊢ (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)))
9	6, 7, 8	mp2an 689	. . . 4 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))
10		rankdmr1 9559	. . . . . 6 ⊢ (rank‘𝐵) ∈ dom 𝑅1
11		rankdmr1 9559	. . . . . 6 ⊢ (rank‘𝐴) ∈ dom 𝑅1
12		r1ord3g 9537	. . . . . 6 ⊢ (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))))
13	10, 11, 12	mp2an 689	. . . . 5 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))
14		r1rankidb 9562	. . . . . 6 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
15	14	sselda 3921	. . . . 5 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐵)))
16		ssel 3914	. . . . 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	syl5bir 242	. . 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 413	1 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ⊆ wss 3887  ∪ cuni 4839  dom cdm 5589   “ cima 5592  Oncon0 6266  ‘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:  wfelirr  9583  rankval3b  9584  rankel  9597  rankunb  9608  rankuni2b  9611  rankcf  10533