Theorem carduni 8807

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		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
2		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
3	1, 2	eqeq12d 2637	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
4	3	rspcv 3305	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘𝑦) = 𝑦))
5		cardon 8770	. . . . . . . . 9 ⊢ (card‘𝑦) ∈ On
6		eleq1 2689	. . . . . . . . 9 ⊢ ((card‘𝑦) = 𝑦 → ((card‘𝑦) ∈ On ↔ 𝑦 ∈ On))
7	5, 6	mpbii 223	. . . . . . . 8 ⊢ ((card‘𝑦) = 𝑦 → 𝑦 ∈ On)
8	4, 7	syl6com 37	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
9	8	ssrdv 3609	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → 𝐴 ⊆ On)
10		ssonuni 6986	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
11	9, 10	syl5 34	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → ∪ 𝐴 ∈ On))
12	11	imp 445	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ∪ 𝐴 ∈ On)
13		cardonle 8783	. . . 4 ⊢ (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ⊆ ∪ 𝐴)
14	12, 13	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → (card‘∪ 𝐴) ⊆ ∪ 𝐴)
15		cardon 8770	. . . . 5 ⊢ (card‘∪ 𝐴) ∈ On
16	15	onirri 5834	. . . 4 ⊢ ¬ (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)
17		eluni 4439	. . . . . . . 8 ⊢ ((card‘∪ 𝐴) ∈ ∪ 𝐴 ↔ ∃𝑦((card‘∪ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
18		elssuni 4467	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
19		ssdomg 8001	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝐴 ∈ On → (𝑦 ⊆ ∪ 𝐴 → 𝑦 ≼ ∪ 𝐴))
20	19	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ⊆ ∪ 𝐴 → 𝑦 ≼ ∪ 𝐴))
21	18, 20	syl5 34	. . . . . . . . . . . . . . . . 17 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → 𝑦 ≼ ∪ 𝐴))
22		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘𝑦) = 𝑦 → (card‘𝑦) = 𝑦)
23		onenon 8775	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((card‘𝑦) ∈ On → (card‘𝑦) ∈ dom card)
24	5, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (card‘𝑦) ∈ dom card
25	22, 24	syl6eqelr 2710	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘𝑦) = 𝑦 → 𝑦 ∈ dom card)
26		onenon 8775	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 ∈ dom card)
27		carddom2 8803	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ dom card ∧ ∪ 𝐴 ∈ dom card) → ((card‘𝑦) ⊆ (card‘∪ 𝐴) ↔ 𝑦 ≼ ∪ 𝐴))
28	25, 26, 27	syl2an 494	. . . . . . . . . . . . . . . . 17 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘∪ 𝐴) ↔ 𝑦 ≼ ∪ 𝐴))
29	21, 28	sylibrd 249	. . . . . . . . . . . . . . . 16 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → (card‘𝑦) ⊆ (card‘∪ 𝐴)))
30		sseq1 3626	. . . . . . . . . . . . . . . . 17 ⊢ ((card‘𝑦) = 𝑦 → ((card‘𝑦) ⊆ (card‘∪ 𝐴) ↔ 𝑦 ⊆ (card‘∪ 𝐴)))
31	30	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → ((card‘𝑦) ⊆ (card‘∪ 𝐴) ↔ 𝑦 ⊆ (card‘∪ 𝐴)))
32	29, 31	sylibd 229	. . . . . . . . . . . . . . 15 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → 𝑦 ⊆ (card‘∪ 𝐴)))
33		ssel 3597	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ (card‘∪ 𝐴) → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))
34	32, 33	syl6 35	. . . . . . . . . . . . . 14 ⊢ (((card‘𝑦) = 𝑦 ∧ ∪ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
35	34	ex 450	. . . . . . . . . . . . 13 ⊢ ((card‘𝑦) = 𝑦 → (∪ 𝐴 ∈ On → (𝑦 ∈ 𝐴 → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))))
36	35	com3r 87	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → ((card‘𝑦) = 𝑦 → (∪ 𝐴 ∈ On → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))))
37	4, 36	syld 47	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → ((card‘∪ 𝐴) ∈ 𝑦 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))))
38	37	com4r 94	. . . . . . . . . 10 ⊢ ((card‘∪ 𝐴) ∈ 𝑦 → (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))))
39	38	imp 445	. . . . . . . . 9 ⊢ (((card‘∪ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
40	39	exlimiv 1858	. . . . . . . 8 ⊢ (∃𝑦((card‘∪ 𝐴) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
41	17, 40	sylbi 207	. . . . . . 7 ⊢ ((card‘∪ 𝐴) ∈ ∪ 𝐴 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (∪ 𝐴 ∈ On → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
42	41	com13 88	. . . . . 6 ⊢ (∪ 𝐴 ∈ On → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → ((card‘∪ 𝐴) ∈ ∪ 𝐴 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴))))
43	42	imp 445	. . . . 5 ⊢ ((∪ 𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ((card‘∪ 𝐴) ∈ ∪ 𝐴 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))
44	12, 43	sylancom 701	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ((card‘∪ 𝐴) ∈ ∪ 𝐴 → (card‘∪ 𝐴) ∈ (card‘∪ 𝐴)))
45	16, 44	mtoi 190	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ¬ (card‘∪ 𝐴) ∈ ∪ 𝐴)
46	15	onordi 5832	. . . 4 ⊢ Ord (card‘∪ 𝐴)
47		eloni 5733	. . . . 5 ⊢ (∪ 𝐴 ∈ On → Ord ∪ 𝐴)
48	12, 47	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → Ord ∪ 𝐴)
49		ordtri4 5761	. . . 4 ⊢ ((Ord (card‘∪ 𝐴) ∧ Ord ∪ 𝐴) → ((card‘∪ 𝐴) = ∪ 𝐴 ↔ ((card‘∪ 𝐴) ⊆ ∪ 𝐴 ∧ ¬ (card‘∪ 𝐴) ∈ ∪ 𝐴)))
50	46, 48, 49	sylancr 695	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → ((card‘∪ 𝐴) = ∪ 𝐴 ↔ ((card‘∪ 𝐴) ⊆ ∪ 𝐴 ∧ ¬ (card‘∪ 𝐴) ∈ ∪ 𝐴)))
51	14, 45, 50	mpbir2and 957	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥) → (card‘∪ 𝐴) = ∪ 𝐴)
52	51	ex 450	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘∪ 𝐴) = ∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114  Ord word 5722  Oncon0 5723  ‘cfv 5888   ≼ cdom 7953  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765

This theorem is referenced by:  cardiun  8808  carduniima  8919