Theorem rankuni 3533

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
rankuni	⊢ (rank ‘∪A) = ∪x ∈ A (rank ‘x)

Distinct variable group(s):   x,A

Proof of Theorem rankuni

Step	Hyp	Ref	Expression
1		ranksn.1	. . . . 5 ⊢ A ∈ V
2	1	uniex 1947	. . . 4 ⊢ ∪A ∈ V
3	2	rankval3 3525	. . 3 ⊢ (rank ‘∪A) = ∩{z ∈ On∣∀y ∈ ∪ A(rank ‘y) ∈ z}
4		fvex 2838	. . . . . . . 8 ⊢ (rank ‘x) ∈ V
5	1, 4	iunon 2947	. . . . . . 7 ⊢ (∀x ∈ A (rank ‘x) ∈ On → ∪x ∈ A (rank ‘x) ∈ On)
6		rankon 3515	. . . . . . . 8 ⊢ (rank ‘x) ∈ On
7	6	a1i 7	. . . . . . 7 ⊢ (x ∈ A → (rank ‘x) ∈ On)
8	5, 7	mprg 1249	. . . . . 6 ⊢ ∪x ∈ A (rank ‘x) ∈ On
9		eluni2 1923	. . . . . . . 8 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
10		ax-17 925	. . . . . . . . . 10 ⊢ (z ∈ (rank ‘y) → ∀x z ∈ (rank ‘y))
11		hbiu1 2012	. . . . . . . . . 10 ⊢ (z ∈ ∪x ∈ A (rank ‘x) → ∀x z ∈ ∪x ∈ A (rank ‘x))
12	10, 11	hbel 1172	. . . . . . . . 9 ⊢ ((rank ‘y) ∈ ∪x ∈ A (rank ‘x) → ∀x(rank ‘y) ∈ ∪x ∈ A (rank ‘x))
13		ssiun2 2019	. . . . . . . . . . 11 ⊢ (x ∈ A → (rank ‘x) ⊆ ∪x ∈ A (rank ‘x))
14	13	sseld 1506	. . . . . . . . . 10 ⊢ (x ∈ A → ((rank ‘y) ∈ (rank ‘x) → (rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
15		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
16	15	rankel 3524	. . . . . . . . . 10 ⊢ (y ∈ x → (rank ‘y) ∈ (rank ‘x))
17	14, 16	syl5 22	. . . . . . . . 9 ⊢ (x ∈ A → (y ∈ x → (rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
18	12, 17	r19.23ai 1283	. . . . . . . 8 ⊢ (∃x ∈ A y ∈ x → (rank ‘y) ∈ ∪x ∈ A (rank ‘x))
19	9, 18	sylbi 174	. . . . . . 7 ⊢ (y ∈ ∪A → (rank ‘y) ∈ ∪x ∈ A (rank ‘x))
20	19	rgen 1247	. . . . . 6 ⊢ ∀y ∈ ∪ A(rank ‘y) ∈ ∪x ∈ A (rank ‘x)
21	8, 20	pm3.2i 234	. . . . 5 ⊢ (∪x ∈ A (rank ‘x) ∈ On ∧ ∀y ∈ ∪ A(rank ‘y) ∈ ∪x ∈ A (rank ‘x))
22		eleq2 1150	. . . . . . 7 ⊢ (z = ∪x ∈ A (rank ‘x) → ((rank ‘y) ∈ z ↔ (rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
23	22	biraldv 1219	. . . . . 6 ⊢ (z = ∪x ∈ A (rank ‘x) → (∀y ∈ ∪ A(rank ‘y) ∈ z ↔ ∀y ∈ ∪ A(rank ‘y) ∈ ∪x ∈ A (rank ‘x)))
24	23	elrab 1422	. . . . 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)))
25	21, 24	mpbir 165	. . . 4 ⊢ ∪x ∈ A (rank ‘x) ∈ {z ∈ On∣∀y ∈ ∪ A(rank ‘y) ∈ z}
26		intss1 1979	. . . 4 ⊢ (∪x ∈ A (rank ‘x) ∈ {z ∈ On∣∀y ∈ ∪ A(rank ‘y) ∈ z} → ∩{z ∈ On∣∀y ∈ ∪ A(rank ‘y) ∈ z} ⊆ ∪x ∈ A (rank ‘x))
27	25, 26	ax-mp 6	. . 3 ⊢ ∩{z ∈ On∣∀y ∈ ∪ A(rank ‘y) ∈ z} ⊆ ∪x ∈ A (rank ‘x)
28	3, 27	eqsstr 1530	. 2 ⊢ (rank ‘∪A) ⊆ ∪x ∈ A (rank ‘x)
29		iunss 2017	. . 3 ⊢ (∪x ∈ A (rank ‘x) ⊆ (rank ‘∪A) ↔ ∀x ∈ A (rank ‘x) ⊆ (rank ‘∪A))
30		elssuni 1940	. . . 4 ⊢ (x ∈ A → x ⊆ ∪A)
31	2	rankss 3531	. . . 4 ⊢ (x ⊆ ∪A → (rank ‘x) ⊆ (rank ‘∪A))
32	30, 31	syl 12	. . 3 ⊢ (x ∈ A → (rank ‘x) ⊆ (rank ‘∪A))
33	29, 32	mprgbir 1250	. 2 ⊢ ∪x ∈ A (rank ‘x) ⊆ (rank ‘∪A)
34	28, 33	eqssi 1517	1 ⊢ (rank ‘∪A) = ∪x ∈ A (rank ‘x)

Syntax hints:   ∧ wa 196   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∪cuni 1919  ∩cint 1965  ∪ciun 1994  Oncon0 2199   ‘cfv 2422  rankcrnk 3486

This theorem is referenced by:  rankuniss 3534

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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