Theorem rankelpr 9679

Description: Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Hypotheses

Ref	Expression
rankelun.1	⊢ 𝐴 ∈ V
rankelun.2	⊢ 𝐵 ∈ V
rankelun.3	⊢ 𝐶 ∈ V
rankelun.4	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
rankelpr	⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))

Proof of Theorem rankelpr

Step	Hyp	Ref	Expression
1		rankelun.1	. . . . 5 ⊢ 𝐴 ∈ V
2		rankelun.2	. . . . 5 ⊢ 𝐵 ∈ V
3		rankelun.3	. . . . 5 ⊢ 𝐶 ∈ V
4		rankelun.4	. . . . 5 ⊢ 𝐷 ∈ V
5	1, 2, 3, 4	rankelun 9678	. . . 4 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷)))
6	1, 2	rankun 9662	. . . 4 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
7	3, 4	rankun 9662	. . . 4 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
8	5, 6, 7	3eltr3g 2853	. . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
9		rankon 9601	. . . . . 6 ⊢ (rank‘𝐶) ∈ On
10		rankon 9601	. . . . . 6 ⊢ (rank‘𝐷) ∈ On
11	9, 10	onun2i 6401	. . . . 5 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
12	11	onordi 6390	. . . 4 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷))
13		ordsucelsuc 7701	. . . 4 ⊢ (Ord ((rank‘𝐶) ∪ (rank‘𝐷)) → (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)) ↔ suc ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ suc ((rank‘𝐶) ∪ (rank‘𝐷))))
14	12, 13	ax-mp 5	. . 3 ⊢ (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)) ↔ suc ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ suc ((rank‘𝐶) ∪ (rank‘𝐷)))
15	8, 14	sylib 217	. 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ suc ((rank‘𝐶) ∪ (rank‘𝐷)))
16	1, 2	rankpr 9663	. 2 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))
17	3, 4	rankpr 9663	. 2 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷))
18	15, 16, 17	3eltr4g 2854	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2104  Vcvv 3437   ∪ cun 3890  {cpr 4567  Ord word 6280  suc csuc 6283  ‘cfv 6458  rankcrnk 9569

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-reg 9399  ax-inf2 9447

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-r1 9570  df-rank 9571

This theorem is referenced by:  rankelop  9680