Theorem rankelpr 9845

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 9844	. . . 4 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷)))
6	1, 2	rankun 9828	. . . 4 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
7	3, 4	rankun 9828	. . . 4 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
8	5, 6, 7	3eltr3g 2885	. . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
9		rankon 9767	. . . . . 6 ⊢ (rank‘𝐶) ∈ On
10		rankon 9767	. . . . . 6 ⊢ (rank‘𝐷) ∈ On
11	9, 10	onun2i 6485	. . . . 5 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
12	11	onordi 6475	. . . 4 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷))
13		ordsucelsuc 7818	. . . 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 221	. 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ suc ((rank‘𝐶) ∪ (rank‘𝐷)))
16	1, 2	rankpr 9829	. 2 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))
17	3, 4	rankpr 9829	. 2 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷))
18	15, 16, 17	3eltr4g 2886	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  Vcvv 3461   ∪ cun 3909  {cpr 4594  Ord word 6360  suc csuc 6363  ‘cfv 6537  rankcrnk 9735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-reg 9554  ax-inf2 9610

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-r1 9736  df-rank 9737

This theorem is referenced by:  rankelop  9846