Theorem rankelun 9825

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 9748	. . . . 5 ⊢ (rank‘𝐶) ∈ On
2		rankon 9748	. . . . 5 ⊢ (rank‘𝐷) ∈ On
3	1, 2	onun2i 6456	. . . 4 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
4	3	onordi 6445	. . 3 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷))
5		elun1 4145	. . 3 ⊢ ((rank‘𝐴) ∈ (rank‘𝐶) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
6		elun2 4146	. . 3 ⊢ ((rank‘𝐵) ∈ (rank‘𝐷) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
7		ordunel 7802	. . 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 9809	. 2 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
12		rankelun.3	. . 3 ⊢ 𝐶 ∈ V
13		rankelun.4	. . 3 ⊢ 𝐷 ∈ V
14	12, 13	rankun 9809	. 2 ⊢ (rank‘(𝐶 ∪ 𝐷)) = ((rank‘𝐶) ∪ (rank‘𝐷))
15	8, 11, 14	3eltr4g 2845	1 ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912  Ord word 6331  ‘cfv 6511  rankcrnk 9716

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-reg 9545  ax-inf2 9594

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-r1 9717  df-rank 9718

This theorem is referenced by:  rankelpr  9826  rankxplim  9832