Theorem carduni 9934

Description: The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.)

Assertion

Ref	Expression
carduni	⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘∪ 𝐴) = ∪ 𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem carduni

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssonuni 7756	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
2		fveq2 6858	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
3		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	2, 3	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
5	4	rspcv 3584	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘𝑦) = 𝑦))
6		cardon 9897	. . . . . . . 8 ⊢ (card‘𝑦) ∈ On
7		eleq1 2816	. . . . . . . 8 ⊢ ((card‘𝑦) = 𝑦 → ((card‘𝑦) ∈ On ↔ 𝑦 ∈ On))
8	6, 7	mpbii 233	. . . . . . 7 ⊢ ((card‘𝑦) = 𝑦 → 𝑦 ∈ On)
9	5, 8	syl6com 37	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
10	9	ssrdv 3952	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → 𝐴 ⊆ On)
11	1, 10	impel 505	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ∪ 𝐴 ∈ On)
12		cardonle 9910	. . . 4 ⊢ (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ⊆ ∪ 𝐴)
13	11, 12	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → (card‘∪ 𝐴) ⊆ ∪ 𝐴)
14		cardon 9897	. . . . 5 ⊢ (card‘∪ 𝐴) ∈ On
15	14	onirri 6447	. . . 4 ⊢ ¬ (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)
16		eluni 4874	. . . . . . . 8 ⊢ ((card‘∪ 𝐴) ∈ ∪ 𝐴 ↔ ∃𝑦((card‘∪ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
17		elssuni 4901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
18		ssdomg 8971	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝐴 ∈ On → (𝑦 ⊆ ∪ 𝐴 → 𝑦 ≼ ∪ 𝐴))
19	18	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ⊆ ∪ 𝐴 → 𝑦 ≼ ∪ 𝐴))
20	17, 19	syl5 34	. . . . . . . . . . . . . . . . 17 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → 𝑦 ≼ ∪ 𝐴))
21		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘𝑦) = 𝑦 → (card‘𝑦) = 𝑦)
22		onenon 9902	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((card‘𝑦) ∈ On → (card‘𝑦) ∈ dom card)
23	6, 22	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (card‘𝑦) ∈ dom card
24	21, 23	eqeltrrdi 2837	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘𝑦) = 𝑦 → 𝑦 ∈ dom card)
25		onenon 9902	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 ∈ dom card)
26		carddom2 9930	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ dom card ∧ ∪ 𝐴 ∈ dom card) → ((card‘𝑦) ⊆ (card‘∪ 𝐴) ↔ 𝑦 ≼ ∪ 𝐴))
27	24, 25, 26	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘∪ 𝐴) ↔ 𝑦 ≼ ∪ 𝐴))
28	20, 27	sylibrd 259	. . . . . . . . . . . . . . . 16 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → (card‘𝑦) ⊆ (card‘∪ 𝐴)))
29		sseq1 3972	. . . . . . . . . . . . . . . . 17 ⊢ ((card‘𝑦) = 𝑦 → ((card‘𝑦) ⊆ (card‘∪ 𝐴) ↔ 𝑦 ⊆ (card‘∪ 𝐴)))
30	29	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘∪ 𝐴) ↔ 𝑦 ⊆ (card‘∪ 𝐴)))
31	28, 30	sylibd 239	. . . . . . . . . . . . . . 15 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → 𝑦 ⊆ (card‘∪ 𝐴)))
32		ssel 3940	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ (card‘∪ 𝐴) → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))
33	31, 32	syl6 35	. . . . . . . . . . . . . 14 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
34	33	ex 412	. . . . . . . . . . . . 13 ⊢ ((card‘𝑦) = 𝑦 → (∪ 𝐴 ∈ On → (𝑦 ∈ 𝐴 → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))))
35	34	com3r 87	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → ((card‘𝑦) = 𝑦 → (∪ 𝐴 ∈ On → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))))
36	5, 35	syld 47	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))))
37	36	com4r 94	. . . . . . . . . 10 ⊢ ((card‘∪ 𝐴) ∈ 𝑦 → (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))))
38	37	imp 406	. . . . . . . . 9 ⊢ (((card‘∪ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
39	38	exlimiv 1930	. . . . . . . 8 ⊢ (∃𝑦((card‘∪ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
40	16, 39	sylbi 217	. . . . . . 7 ⊢ ((card‘∪ 𝐴) ∈ ∪ 𝐴 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
41	40	com13 88	. . . . . 6 ⊢ (∪ 𝐴 ∈ On → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → ((card‘∪ 𝐴) ∈ ∪ 𝐴 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
42	41	imp 406	. . . . 5 ⊢ ((∪ 𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ((card‘∪ 𝐴) ∈ ∪ 𝐴 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))
43	11, 42	sylancom 588	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ((card‘∪ 𝐴) ∈ ∪ 𝐴 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))
44	15, 43	mtoi 199	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ¬ (card‘∪ 𝐴) ∈ ∪ 𝐴)
45	14	onordi 6445	. . . 4 ⊢ Ord (card‘∪ 𝐴)
46		eloni 6342	. . . . 5 ⊢ (∪ 𝐴 ∈ On → Ord ∪ 𝐴)
47	11, 46	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → Ord ∪ 𝐴)
48		ordtri4 6369	. . . 4 ⊢ ((Ord (card‘∪ 𝐴) ∧ Ord ∪ 𝐴) → ((card‘∪ 𝐴) = ∪ 𝐴 ↔ ((card‘∪ 𝐴) ⊆ ∪ 𝐴 ∧ ¬ (card‘∪ 𝐴) ∈ ∪ 𝐴)))
49	45, 47, 48	sylancr 587	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ((card‘∪ 𝐴) = ∪ 𝐴 ↔ ((card‘∪ 𝐴) ⊆ ∪ 𝐴 ∧ ¬ (card‘∪ 𝐴) ∈ ∪ 𝐴)))
50	13, 44, 49	mpbir2and 713	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → (card‘∪ 𝐴) = ∪ 𝐴)
51	50	ex 412	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘∪ 𝐴) = ∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3914  ∪ cuni 4871   class class class wbr 5107  dom cdm 5638  Ord word 6331  Oncon0 6332  ‘cfv 6511   ≼ cdom 8916  cardccrd 9888

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-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-rab 3406  df-v 3449  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-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-ord 6335  df-on 6336  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-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-card 9892

This theorem is referenced by:  cardiun  9935  carduniima  10049