Theorem rankuni2 4700

Description: The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112.

Hypothesis

Ref	Expression
ranksn.1	⊢ A ∈ V

Assertion

Ref	Expression
rankuni2	⊢ (rank ‘∪A) = ∪x ∈ A (rank ‘x)

Distinct variable group:   x,A

Proof of Theorem rankuni2

Step	Hyp	Ref	Expression
1		ranksn.1	. . . . 5 ⊢ A ∈ V
2	1	uniex 2876	. . . 4 ⊢ ∪A ∈ V
3	2	rankval3 4691	. . 3 ⊢ (rank ‘∪A) = ∩{z ∈ On∣∀y ∈ ∪A(rank ‘y) ∈ z}
4		eleq2 1538	. . . . . . 7 ⊢ (z = ∪x ∈ A (rank ‘x) → ((rank ‘y) ∈ z ↔ (rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
5	4	ralbidv 1666	. . . . . 6 ⊢ (z = ∪x ∈ A (rank ‘x) → (∀y ∈ ∪A(rank ‘y) ∈ z ↔ ∀y ∈ ∪A(rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
6	5	elrab 1908	. . . . 5 ⊢ (∪x ∈ A (rank ‘x) ∈ {z ∈ On∣∀y ∈ ∪A(rank ‘y) ∈ z} ↔ (∪x ∈ A (rank ‘x) ∈ On ⋀ ∀y ∈ ∪A(rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
7		fvex 3738	. . . . . . 7 ⊢ (rank ‘x) ∈ V
8	1, 7	iunon 3915	. . . . . 6 ⊢ (∀x ∈ A (rank ‘x) ∈ On → ∪x ∈ A (rank ‘x) ∈ On)
9		rankon 4681	. . . . . . 7 ⊢ (rank ‘x) ∈ On
10	9	a1i 8	. . . . . 6 ⊢ (x ∈ A → (rank ‘x) ∈ On)
11	8, 10	mprg 1703	. . . . 5 ⊢ ∪x ∈ A (rank ‘x) ∈ On
12		eluni2 2511	. . . . . . 7 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
13		ax-17 973	. . . . . . . . 9 ⊢ (z ∈ (rank ‘y) → ∀x z ∈ (rank ‘y))
14		hbiu1 2588	. . . . . . . . 9 ⊢ (z ∈ ∪x ∈ A (rank ‘x) → ∀x z ∈ ∪x ∈ A (rank ‘x))
15	13, 14	hbel 1569	. . . . . . . 8 ⊢ ((rank ‘y) ∈ ∪x ∈ A (rank ‘x) → ∀x(rank ‘y) ∈ ∪x ∈ A (rank ‘x))
16		ssiun2 2597	. . . . . . . . . 10 ⊢ (x ∈ A → (rank ‘x) ⊆ ∪x ∈ A (rank ‘x))
17	16	sseld 2070	. . . . . . . . 9 ⊢ (x ∈ A → ((rank ‘y) ∈ (rank ‘x) → (rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
18		visset 1816	. . . . . . . . . 10 ⊢ x ∈ V
19	18	rankel 4690	. . . . . . . . 9 ⊢ (y ∈ x → (rank ‘y) ∈ (rank ‘x))
20	17, 19	syl5 21	. . . . . . . 8 ⊢ (x ∈ A → (y ∈ x → (rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
21	15, 20	r19.23ai 1745	. . . . . . 7 ⊢ (∃x ∈ A y ∈ x → (rank ‘y) ∈ ∪x ∈ A (rank ‘x))
22	12, 21	sylbi 199	. . . . . 6 ⊢ (y ∈ ∪A → (rank ‘y) ∈ ∪x ∈ A (rank ‘x))
23	22	rgen 1701	. . . . 5 ⊢ ∀y ∈ ∪A(rank ‘y) ∈ ∪x ∈ A (rank ‘x)
24	6, 11, 23	mpbir2an 732	. . . 4 ⊢ ∪x ∈ A (rank ‘x) ∈ {z ∈ On∣∀y ∈ ∪A(rank ‘y) ∈ z}
25		intss1 2552	. . . 4 ⊢ (∪x ∈ A (rank ‘x) ∈ {z ∈ On∣∀y ∈ ∪A(rank ‘y) ∈ z} → ∩{z ∈ On∣∀y ∈ ∪A(rank ‘y) ∈ z} ⊆ ∪x ∈ A (rank ‘x))
26	24, 25	ax-mp 7	. . 3 ⊢ ∩{z ∈ On∣∀y ∈ ∪A(rank ‘y) ∈ z} ⊆ ∪x ∈ A (rank ‘x)
27	3, 26	eqsstr 2094	. 2 ⊢ (rank ‘∪A) ⊆ ∪x ∈ A (rank ‘x)
28		iunss 2595	. . 3 ⊢ (∪x ∈ A (rank ‘x) ⊆ (rank ‘∪A) ↔ ∀x ∈ A (rank ‘x) ⊆ (rank ‘∪A))
29		elssuni 2530	. . . 4 ⊢ (x ∈ A → x ⊆ ∪A)
30	2	rankss 4698	. . . 4 ⊢ (x ⊆ ∪A → (rank ‘x) ⊆ (rank ‘∪A))
31	29, 30	syl 10	. . 3 ⊢ (x ∈ A → (rank ‘x) ⊆ (rank ‘∪A))
32	28, 31	mprgbir 1704	. 2 ⊢ ∪x ∈ A (rank ‘x) ⊆ (rank ‘∪A)
33	27, 32	eqssi 2081	1 ⊢ (rank ‘∪A) = ∪x ∈ A (rank ‘x)

Syntax hints:   = wceq 958   ∈ wcel 960  ∀wral 1648  ∃wrex 1649  {crab 1651  Vcvv 1814   ⊆ wss 2050  ∪cuni 2507  ∩cint 2537  ∪ciun 2570  Oncon0 2954   ‘cfv 3188  rankcrnk 4652

This theorem is referenced by:  rankuni 4708  rankbnd2 4714

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

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