Theorem rankelb 9794

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 9775	. . . . . 6 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ ∪ (𝑅1 “ On))
2	1	sseld 3935	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ (𝑅1 “ On)))
3		rankidn 9792	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
4	2, 3	syl6 36	. . . 4 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
5	4	imp 411	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
6		rankon 9765	. . . . 5 ⊢ (rank‘𝐵) ∈ On
7		rankon 9765	. . . . 5 ⊢ (rank‘𝐴) ∈ On
8		ontri1 6395	. . . . 5 ⊢ (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)))
9	6, 7, 8	mp2an 704	. . . 4 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))
10		rankdmr1 9771	. . . . . 6 ⊢ (rank‘𝐵) ∈ dom 𝑅1
11		rankdmr1 9771	. . . . . 6 ⊢ (rank‘𝐴) ∈ dom 𝑅1
12		r1ord3g 9749	. . . . . 6 ⊢ (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))))
13	10, 11, 12	mp2an 704	. . . . 5 ⊢ ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))
14		r1rankidb 9774	. . . . . 6 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
15	14	sselda 3936	. . . . 5 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐵)))
16		ssel 3930	. . . . 5 ⊢ ((𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)) → (𝐴 ∈ (𝑅1‘(rank‘𝐵)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
17	13, 15, 16	syl2imc 42	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → ((rank‘𝐵) ⊆ (rank‘𝐴) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
18	9, 17	biimtrrid 246	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → (¬ (rank‘𝐴) ∈ (rank‘𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
19	5, 18	mt3d 149	. 2 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ 𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
20	19	ex 417	1 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2142   ⊆ wss 3904  ∪ cuni 4871  dom cdm 5660   “ cima 5663  Oncon0 6360  ‘cfv 6536  𝑅₁cr1 9732  rankcrnk 9733

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-r1 9734  df-rank 9735

This theorem is used by:  wfelirr  9795  rankval3b  9796  rankel  9809  rankunb  9820  rankuni2b  9823  rankcf  10768  r1elcl  35500  rankval4b  35502  rankfilimb  35505  regsfromunir1  37079  rankrelp  45697