Theorem alephadd 10564

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 6897	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
2		fvex 6897	. . . . 5 ⊢ (ℵ‘𝐵) ∈ V
3		djuex 9896	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V)
4	1, 2, 3	mp2an 704	. . . 4 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V
5		alephfnon 10051	. . . . . . . 8 ⊢ ℵ Fn On
6	5	fndmi 6642	. . . . . . 7 ⊢ dom ℵ = On
7	6	eleq2i 2861	. . . . . 6 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
8	7	notbii 323	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
9	6	eleq2i 2861	. . . . . 6 ⊢ (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
10	9	notbii 323	. . . . 5 ⊢ (¬ 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
11		df-dju 9889	. . . . . . 7 ⊢ (∅ ⊔ ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
12		xpundir 5734	. . . . . . 7 ⊢ (({∅} ∪ {1o}) × ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
13		xp0 5764	. . . . . . 7 ⊢ (({∅} ∪ {1o}) × ∅) = ∅
14	11, 12, 13	3eqtr2i 2798	. . . . . 6 ⊢ (∅ ⊔ ∅) = ∅
15		ndmfv 6916	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16		ndmfv 6916	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
17		djueq12 9892	. . . . . . 7 ⊢ (((ℵ‘𝐴) = ∅ ∧ (ℵ‘𝐵) = ∅) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
18	15, 16, 17	syl2an 607	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
19	15	adantr 485	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
20	16	adantl 486	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
21	19, 20	uneq12d 4131	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
22		un0 4358	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
23	21, 22	eqtrdi 2820	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
24	14, 18, 23	3eqtr4a 2830	. . . . 5 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
25	8, 10, 24	syl2anbr 610	. . . 4 ⊢ ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
26		eqeng 8985	. . . 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 417	. 2 ⊢ (¬ 𝐴 ∈ On → (¬ 𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
29		alephgeom 10068	. . 3 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
30		ssdomg 8999	. . . . 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 10055	. . . . . 6 ⊢ (ℵ‘𝐴) ∈ On
33		onenon 9937	. . . . . 6 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
34	32, 33	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐴) ∈ dom card
35		alephon 10055	. . . . . 6 ⊢ (ℵ‘𝐵) ∈ On
36		onenon 9937	. . . . . 6 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
37	35, 36	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐵) ∈ dom card
38		infdju 10192	. . . . 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 1477	. . . 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 10068	. . 3 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43		ssdomg 8999	. . . . 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 10163	. . . . . . 7 ⊢ (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)))
46	1, 2, 45	mp2an 704	. . . . . 6 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴))
47		infdju 10192	. . . . . . 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 1477	. . . . . 6 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49		entr 9005	. . . . . 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 598	. . . . 5 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51		uncom 4120	. . . . 5 ⊢ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
52	50, 51	breqtrdi 5156	. . . 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))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  {csn 4594   class class class wbr 5113   × cxp 5662  dom cdm 5664  Oncon0 6363  ‘cfv 6539  ωcom 7864  1_oc1o 8448   ≈ cen 8942   ≼ cdom 8943   ⊔ cdju 9886  cardccrd 9923  ℵcale 9924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-inf2 9612

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-oi 9474  df-har 9521  df-dju 9889  df-card 9927  df-aleph 9928

This theorem is referenced by: (None)