Theorem rankelun 4717

Description: Rank membership is inherited by union.

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		elun1 2200	. . . . . . . 8 ⊢ (A ∈ (R1 ‘(rank ‘C)) → A ∈ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))))
2		elun2 2201	. . . . . . . 8 ⊢ (B ∈ (R1 ‘(rank ‘D)) → B ∈ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))))
3	1, 2	anim12i 333	. . . . . . 7 ⊢ ((A ∈ (R1 ‘(rank ‘C)) ⋀ B ∈ (R1 ‘(rank ‘D))) → (A ∈ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))) ⋀ B ∈ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))))
4		rankelun.1	. . . . . . . 8 ⊢ A ∈ V
5		rankelun.2	. . . . . . . 8 ⊢ B ∈ V
6	4, 5	prss 2475	. . . . . . 7 ⊢ ((A ∈ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))) ⋀ B ∈ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))) ↔ {A, B} ⊆ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))))
7	3, 6	sylib 198	. . . . . 6 ⊢ ((A ∈ (R1 ‘(rank ‘C)) ⋀ B ∈ (R1 ‘(rank ‘D))) → {A, B} ⊆ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))))
8		fvex 3738	. . . . . . . 8 ⊢ (R1 ‘(rank ‘C)) ∈ V
9		fvex 3738	. . . . . . . 8 ⊢ (R1 ‘(rank ‘D)) ∈ V
10	8, 9	unex 2878	. . . . . . 7 ⊢ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))) ∈ V
11	10	rankss 4698	. . . . . 6 ⊢ ({A, B} ⊆ ((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))) → (rank ‘{A, B}) ⊆ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))))
12	7, 11	syl 10	. . . . 5 ⊢ ((A ∈ (R1 ‘(rank ‘C)) ⋀ B ∈ (R1 ‘(rank ‘D))) → (rank ‘{A, B}) ⊆ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))))
13	4, 5	rankpr 4702	. . . . . 6 ⊢ (rank ‘{A, B}) = suc ((rank ‘A) ∪ (rank ‘B))
14	4, 5	rankun 4701	. . . . . . 7 ⊢ (rank ‘(A ∪ B)) = ((rank ‘A) ∪ (rank ‘B))
15		suceq 3040	. . . . . . 7 ⊢ ((rank ‘(A ∪ B)) = ((rank ‘A) ∪ (rank ‘B)) → suc (rank ‘(A ∪ B)) = suc ((rank ‘A) ∪ (rank ‘B)))
16	14, 15	ax-mp 7	. . . . . 6 ⊢ suc (rank ‘(A ∪ B)) = suc ((rank ‘A) ∪ (rank ‘B))
17	13, 16	eqtr4 1501	. . . . 5 ⊢ (rank ‘{A, B}) = suc (rank ‘(A ∪ B))
18	12, 17	syl5ssr 2109	. . . 4 ⊢ ((A ∈ (R1 ‘(rank ‘C)) ⋀ B ∈ (R1 ‘(rank ‘D))) → suc (rank ‘(A ∪ B)) ⊆ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))))
19		fvex 3738	. . . . 5 ⊢ (rank ‘(A ∪ B)) ∈ V
20		sucssel 3076	. . . . 5 ⊢ ((rank ‘(A ∪ B)) ∈ V → (suc (rank ‘(A ∪ B)) ⊆ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))) → (rank ‘(A ∪ B)) ∈ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D))))))
21	19, 20	ax-mp 7	. . . 4 ⊢ (suc (rank ‘(A ∪ B)) ⊆ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))) → (rank ‘(A ∪ B)) ∈ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))))
22	18, 21	syl 10	. . 3 ⊢ ((A ∈ (R1 ‘(rank ‘C)) ⋀ B ∈ (R1 ‘(rank ‘D))) → (rank ‘(A ∪ B)) ∈ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))))
23		rankon 4681	. . . 4 ⊢ (rank ‘C) ∈ On
24	4	rankr1a 4687	. . . 4 ⊢ ((rank ‘C) ∈ On → (A ∈ (R1 ‘(rank ‘C)) ↔ (rank ‘A) ∈ (rank ‘C)))
25	23, 24	ax-mp 7	. . 3 ⊢ (A ∈ (R1 ‘(rank ‘C)) ↔ (rank ‘A) ∈ (rank ‘C))
26		rankon 4681	. . . 4 ⊢ (rank ‘D) ∈ On
27	5	rankr1a 4687	. . . 4 ⊢ ((rank ‘D) ∈ On → (B ∈ (R1 ‘(rank ‘D)) ↔ (rank ‘B) ∈ (rank ‘D)))
28	26, 27	ax-mp 7	. . 3 ⊢ (B ∈ (R1 ‘(rank ‘D)) ↔ (rank ‘B) ∈ (rank ‘D))
29	22, 25, 28	syl2anbr 458	. 2 ⊢ (((rank ‘A) ∈ (rank ‘C) ⋀ (rank ‘B) ∈ (rank ‘D)) → (rank ‘(A ∪ B)) ∈ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))))
30		rankr1id 4707	. . . . 5 ⊢ ((rank ‘C) ∈ On ↔ (rank ‘(R1 ‘(rank ‘C))) = (rank ‘C))
31	23, 30	mpbi 189	. . . 4 ⊢ (rank ‘(R1 ‘(rank ‘C))) = (rank ‘C)
32		rankr1id 4707	. . . . 5 ⊢ ((rank ‘D) ∈ On ↔ (rank ‘(R1 ‘(rank ‘D))) = (rank ‘D))
33	26, 32	mpbi 189	. . . 4 ⊢ (rank ‘(R1 ‘(rank ‘D))) = (rank ‘D)
34	31, 33	uneq12i 2185	. . 3 ⊢ ((rank ‘(R1 ‘(rank ‘C))) ∪ (rank ‘(R1 ‘(rank ‘D)))) = ((rank ‘C) ∪ (rank ‘D))
35	8, 9	rankun 4701	. . 3 ⊢ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))) = ((rank ‘(R1 ‘(rank ‘C))) ∪ (rank ‘(R1 ‘(rank ‘D))))
36		rankelun.3	. . . 4 ⊢ C ∈ V
37		rankelun.4	. . . 4 ⊢ D ∈ V
38	36, 37	rankun 4701	. . 3 ⊢ (rank ‘(C ∪ D)) = ((rank ‘C) ∪ (rank ‘D))
39	34, 35, 38	3eqtr4 1508	. 2 ⊢ (rank ‘((R1 ‘(rank ‘C)) ∪ (R1 ‘(rank ‘D)))) = (rank ‘(C ∪ D))
40	29, 39	syl6eleq 1561	1 ⊢ (((rank ‘A) ∈ (rank ‘C) ⋀ (rank ‘B) ∈ (rank ‘D)) → (rank ‘(A ∪ B)) ∈ (rank ‘(C ∪ D)))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814   ∪ cun 2048   ⊆ wss 2050  {cpr 2414  Oncon0 2954  suc csuc 2956   ‘cfv 3188  R₁cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankelpr 4718  rankxplim 4722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

Copyright terms: Public domain