Theorem alephadd 10611

Description: The sum of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
alephadd	⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))

Proof of Theorem alephadd

Step	Hyp	Ref	Expression
1		fvex 6894	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
2		fvex 6894	. . . . 5 ⊢ (ℵ‘𝐵) ∈ V
3		djuex 9938	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V)
4	1, 2, 3	mp2an 705	. . . 4 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V
5		alephfnon 10093	. . . . . . . 8 ⊢ ℵ Fn On
6	5	fndmi 6639	. . . . . . 7 ⊢ dom ℵ = On
7	6	eleq2i 2852	. . . . . 6 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
8	7	notbii 323	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
9	6	eleq2i 2852	. . . . . 6 ⊢ (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
10	9	notbii 323	. . . . 5 ⊢ (¬ 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
11		df-dju 9931	. . . . . . 7 ⊢ (∅ ⊔ ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
12		xpundir 5725	. . . . . . 7 ⊢ (({∅} ∪ {1o}) × ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
13		xp0 5755	. . . . . . 7 ⊢ (({∅} ∪ {1o}) × ∅) = ∅
14	11, 12, 13	3eqtr2i 2789	. . . . . 6 ⊢ (∅ ⊔ ∅) = ∅
15		ndmfv 6913	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16		ndmfv 6913	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
17		djueq12 9934	. . . . . . 7 ⊢ (((ℵ‘𝐴) = ∅ ∧ (ℵ‘𝐵) = ∅) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
18	15, 16, 17	syl2an 608	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
19	15	adantr 486	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
20	16	adantl 487	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
21	19, 20	uneq12d 4116	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
22		un0 4344	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
23	21, 22	eqtrdi 2811	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
24	14, 18, 23	3eqtr4a 2821	. . . . 5 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
25	8, 10, 24	syl2anbr 611	. . . 4 ⊢ ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
26		eqeng 8999	. . . 4 ⊢ (((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V → (((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
27	4, 25, 26	mpsyl 69	. . 3 ⊢ ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
28	27	ex 418	. 2 ⊢ (¬ 𝐴 ∈ On → (¬ 𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
29		alephgeom 10110	. . 3 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
30		ssdomg 9013	. . . . 5 ⊢ ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
31	1, 30	ax-mp 5	. . . 4 ⊢ (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
32		alephon 10097	. . . . . 6 ⊢ (ℵ‘𝐴) ∈ On
33		onenon 9979	. . . . . 6 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
34	32, 33	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐴) ∈ dom card
35		alephon 10097	. . . . . 6 ⊢ (ℵ‘𝐵) ∈ On
36		onenon 9979	. . . . . 6 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
37	35, 36	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐵) ∈ dom card
38		infdju 10234	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ dom card ∧ (ℵ‘𝐵) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
39	34, 37, 38	mp3an12 1480	. . . 4 ⊢ (ω ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
40	31, 39	syl 18	. . 3 ⊢ (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
41	29, 40	sylbi 220	. 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
42		alephgeom 10110	. . 3 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43		ssdomg 9013	. . . . 5 ⊢ ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)))
44	2, 43	ax-mp 5	. . . 4 ⊢ (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))
45		djucomen 10205	. . . . . . 7 ⊢ (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)))
46	1, 2, 45	mp2an 705	. . . . . 6 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴))
47		infdju 10234	. . . . . . 7 ⊢ (((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐵)) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
48	37, 34, 47	mp3an12 1480	. . . . . 6 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49		entr 9019	. . . . . 6 ⊢ ((((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ∧ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴))) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
50	46, 48, 49	sylancr 599	. . . . 5 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51		uncom 4105	. . . . 5 ⊢ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
52	50, 51	breqtrdi 5146	. . . 4 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
53	44, 52	syl 18	. . 3 ⊢ (ω ⊆ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
54	42, 53	sylbi 220	. 2 ⊢ (𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
55	28, 41, 54	pm2.61ii 185	1 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  {csn 4584   class class class wbr 5103   × cxp 5653  dom cdm 5655  Oncon0 6359  ‘cfv 6535  ωcom 7868  1_oc1o 8455   ≈ cen 8956   ≼ cdom 8957   ⊔ cdju 9928  cardccrd 9965  ℵcale 9966

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-inf2 9627

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8703  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-oi 9489  df-har 9536  df-dju 9931  df-card 9969  df-aleph 9970

This theorem is used by: (None)