Theorem rankelun 9801

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 9724	. . . . 5 ⊢ (rank‘𝐶) ∈ On
2		rankon 9724	. . . . 5 ⊢ (rank‘𝐷) ∈ On
3	1, 2	onun2i 6444	. . . 4 ⊢ ((rank‘𝐶) ∪ (rank‘𝐷)) ∈ On
4	3	onordi 6433	. . 3 ⊢ Ord ((rank‘𝐶) ∪ (rank‘𝐷))
5		elun1 4141	. . 3 ⊢ ((rank‘𝐴) ∈ (rank‘𝐶) → (rank‘𝐴) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
6		elun2 4142	. . 3 ⊢ ((rank‘𝐵) ∈ (rank‘𝐷) → (rank‘𝐵) ∈ ((rank‘𝐶) ∪ (rank‘𝐷)))
7		ordunel 7782	. . 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 9785	. 2 ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))
12		rankelun.3	. . 3 ⊢ 𝐶 ∈ V
13		rankelun.4	. . 3 ⊢ 𝐷 ∈ V
14	12, 13	rankun 9785	. 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 3444   ∪ cun 3909  Ord word 6319  ‘cfv 6499  rankcrnk 9692

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-r1 9693  df-rank 9694

This theorem is referenced by:  rankelpr  9802  rankxplim  9808