Theorem rankelun 9870

Description: Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened 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
rankelun	⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷)))

Proof of Theorem rankelun

Step	Hyp	Ref	Expression
1		rankon 9793	. . . . 5 ⊢ (rank‘𝐶) ∈ On
2		rankon 9793	. . . . 5 ⊢ (rank‘𝐷) ∈ On
3	1, 2	onun2i 6486	. . . 4 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
4	3	onordi 6475	. . 3 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷))
5		elun1 4176	. . 3 ⊢ ((rank‘𝐴) ∈ (rank‘𝐶) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
6		elun2 4177	. . 3 ⊢ ((rank‘𝐵) ∈ (rank‘𝐷) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
7		ordunel 7818	. . 3 ⊢ ((Ord ((rank‘𝐶) ∪ (rank‘𝐷)) ∧ (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)) ∧ (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷))) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
8	4, 5, 6, 7	mp3an3an 1466	. 2 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
9		rankelun.1	. . 3 ⊢ 𝐴 ∈ V
10		rankelun.2	. . 3 ⊢ 𝐵 ∈ V
11	9, 10	rankun 9854	. 2 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
12		rankelun.3	. . 3 ⊢ 𝐶 ∈ V
13		rankelun.4	. . 3 ⊢ 𝐷 ∈ V
14	12, 13	rankun 9854	. 2 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
15	8, 11, 14	3eltr4g 2849	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105  Vcvv 3473   ∪ cun 3946  Ord word 6363  ‘cfv 6543  rankcrnk 9761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-reg 9590  ax-inf2 9639

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-r1 9762  df-rank 9763

This theorem is referenced by:  rankelpr  9871  rankxplim  9877