Theorem rankelun 9886

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 9809	. . . . 5 ⊢ (rank‘𝐶) ∈ On
2		rankon 9809	. . . . 5 ⊢ (rank‘𝐷) ∈ On
3	1, 2	onun2i 6476	. . . 4 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
4	3	onordi 6465	. . 3 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷))
5		elun1 4157	. . 3 ⊢ ((rank‘𝐴) ∈ (rank‘𝐶) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
6		elun2 4158	. . 3 ⊢ ((rank‘𝐵) ∈ (rank‘𝐷) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
7		ordunel 7821	. . 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 1469	. 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 9870	. 2 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
12		rankelun.3	. . 3 ⊢ 𝐶 ∈ V
13		rankelun.4	. . 3 ⊢ 𝐷 ∈ V
14	12, 13	rankun 9870	. 2 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
15	8, 11, 14	3eltr4g 2851	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924  Ord word 6351  ‘cfv 6531  rankcrnk 9777

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-reg 9606  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-r1 9778  df-rank 9779

This theorem is referenced by:  rankelpr  9887  rankxplim  9893