Theorem rankelpr 9914

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 9913	. . . 4 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷)))
6	1, 2	rankun 9894	. . . 4 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
7	3, 4	rankun 9894	. . . 4 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
8	5, 6, 7	3eltr3g 2880	. . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
9		rankon 9827	. . . . . 6 ⊢ (rank‘𝐶) ∈ On
10		rankon 9827	. . . . . 6 ⊢ (rank‘𝐷) ∈ On
11	9, 10	onun2i 6501	. . . . 5 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
12	11	onordi 6491	. . . 4 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷))
13		ordsucelsuc 7854	. . . 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 9895	. 2 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))
17	3, 4	rankpr 9895	. 2 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷))
18	15, 16, 17	3eltr4g 2881	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   ∪ cun 4152  {cpr 4590  Ord word 6376  suc csuc 6379  ‘cfv 6555  rankcrnk 9791

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-reg 9610  ax-inf2 9666

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-r1 9792  df-rank 9793

This theorem is used by:  rankelop  9915