Theorem alephadd 10471

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 6835	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
2		fvex 6835	. . . . 5 ⊢ (ℵ‘𝐵) ∈ V
3		djuex 9804	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V)
4	1, 2, 3	mp2an 692	. . . 4 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V
5		alephfnon 9959	. . . . . . . 8 ⊢ ℵ Fn On
6	5	fndmi 6586	. . . . . . 7 ⊢ dom ℵ = On
7	6	eleq2i 2820	. . . . . 6 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
8	7	notbii 320	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
9	6	eleq2i 2820	. . . . . 6 ⊢ (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
10	9	notbii 320	. . . . 5 ⊢ (¬ 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
11		df-dju 9797	. . . . . . 7 ⊢ (∅ ⊔ ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
12		xpundir 5689	. . . . . . 7 ⊢ (({∅} ∪ {1o}) × ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
13		xp0 6107	. . . . . . 7 ⊢ (({∅} ∪ {1o}) × ∅) = ∅
14	11, 12, 13	3eqtr2i 2758	. . . . . 6 ⊢ (∅ ⊔ ∅) = ∅
15		ndmfv 6855	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16		ndmfv 6855	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
17		djueq12 9800	. . . . . . 7 ⊢ (((ℵ‘𝐴) = ∅ ∧ (ℵ‘𝐵) = ∅) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
18	15, 16, 17	syl2an 596	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
19	15	adantr 480	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
20	16	adantl 481	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
21	19, 20	uneq12d 4120	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
22		un0 4345	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
23	21, 22	eqtrdi 2780	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
24	14, 18, 23	3eqtr4a 2790	. . . . 5 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
25	8, 10, 24	syl2anbr 599	. . . 4 ⊢ ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
26		eqeng 8911	. . . 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 68	. . 3 ⊢ ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
28	27	ex 412	. 2 ⊢ (¬ 𝐴 ∈ On → (¬ 𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
29		alephgeom 9976	. . 3 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
30		ssdomg 8925	. . . . 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 9963	. . . . . 6 ⊢ (ℵ‘𝐴) ∈ On
33		onenon 9845	. . . . . 6 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
34	32, 33	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐴) ∈ dom card
35		alephon 9963	. . . . . 6 ⊢ (ℵ‘𝐵) ∈ On
36		onenon 9845	. . . . . 6 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
37	35, 36	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐵) ∈ dom card
38		infdju 10101	. . . . 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 1453	. . . 4 ⊢ (ω ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
40	31, 39	syl 17	. . 3 ⊢ (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
41	29, 40	sylbi 217	. 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
42		alephgeom 9976	. . 3 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43		ssdomg 8925	. . . . 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 10072	. . . . . . 7 ⊢ (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)))
46	1, 2, 45	mp2an 692	. . . . . 6 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴))
47		infdju 10101	. . . . . . 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 1453	. . . . . 6 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49		entr 8931	. . . . . 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 587	. . . . 5 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51		uncom 4109	. . . . 5 ⊢ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
52	50, 51	breqtrdi 5133	. . . 4 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
53	44, 52	syl 17	. . 3 ⊢ (ω ⊆ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
54	42, 53	sylbi 217	. 2 ⊢ (𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
55	28, 41, 54	pm2.61ii 183	1 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  {csn 4577   class class class wbr 5092   × cxp 5617  dom cdm 5619  Oncon0 6307  ‘cfv 6482  ωcom 7799  1_oc1o 8381   ≈ cen 8869   ≼ cdom 8870   ⊔ cdju 9794  cardccrd 9831  ℵcale 9832

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-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

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-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-oi 9402  df-har 9449  df-dju 9797  df-card 9835  df-aleph 9836

This theorem is referenced by: (None)