Theorem rankelpr 9305

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 9304	. . . 4 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷)))
6	1, 2	rankun 9288	. . . 4 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
7	3, 4	rankun 9288	. . . 4 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
8	5, 6, 7	3eltr3g 2932	. . 3 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
9		rankon 9227	. . . . . 6 ⊢ (rank‘𝐶) ∈ On
10		rankon 9227	. . . . . 6 ⊢ (rank‘𝐷) ∈ On
11	9, 10	onun2i 6309	. . . . 5 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
12	11	onordi 6298	. . . 4 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷))
13		ordsucelsuc 7540	. . . 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 220	. 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → suc ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ suc ((rank‘𝐶) ∪ (rank‘𝐷)))
16	1, 2	rankpr 9289	. 2 ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))
17	3, 4	rankpr 9289	. 2 ⊢ (rank‘{𝐶, 𝐷}) = suc ((rank‘𝐶) ∪ (rank‘𝐷))
18	15, 16, 17	3eltr4g 2933	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2113  Vcvv 3497   ∪ cun 3937  {cpr 4572  Ord word 6193  suc csuc 6196  ‘cfv 6358  rankcrnk 9195

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-reg 9059  ax-inf2 9107

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-r1 9196  df-rank 9197

This theorem is referenced by:  rankelop  9306