Theorem alephadd 10264

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 6769	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
2		fvex 6769	. . . . 5 ⊢ (ℵ‘𝐵) ∈ V
3		djuex 9597	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V)
4	1, 2, 3	mp2an 688	. . . 4 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V
5		alephfnon 9752	. . . . . . . 8 ⊢ ℵ Fn On
6	5	fndmi 6521	. . . . . . 7 ⊢ dom ℵ = On
7	6	eleq2i 2830	. . . . . 6 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
8	7	notbii 319	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
9	6	eleq2i 2830	. . . . . 6 ⊢ (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
10	9	notbii 319	. . . . 5 ⊢ (¬ 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
11		df-dju 9590	. . . . . . 7 ⊢ (∅ ⊔ ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
12		xpundir 5647	. . . . . . 7 ⊢ (({∅} ∪ {1o}) × ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
13		xp0 6050	. . . . . . 7 ⊢ (({∅} ∪ {1o}) × ∅) = ∅
14	11, 12, 13	3eqtr2i 2772	. . . . . 6 ⊢ (∅ ⊔ ∅) = ∅
15		ndmfv 6786	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16		ndmfv 6786	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
17		djueq12 9593	. . . . . . 7 ⊢ (((ℵ‘𝐴) = ∅ ∧ (ℵ‘𝐵) = ∅) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
18	15, 16, 17	syl2an 595	. . . . . 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 4094	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
22		un0 4321	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
23	21, 22	eqtrdi 2795	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
24	14, 18, 23	3eqtr4a 2805	. . . . 5 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
25	8, 10, 24	syl2anbr 598	. . . 4 ⊢ ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
26		eqeng 8729	. . . 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 9769	. . 3 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
30		ssdomg 8741	. . . . 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 9756	. . . . . 6 ⊢ (ℵ‘𝐴) ∈ On
33		onenon 9638	. . . . . 6 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
34	32, 33	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐴) ∈ dom card
35		alephon 9756	. . . . . 6 ⊢ (ℵ‘𝐵) ∈ On
36		onenon 9638	. . . . . 6 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
37	35, 36	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐵) ∈ dom card
38		infdju 9895	. . . . 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 1449	. . . 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 216	. 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
42		alephgeom 9769	. . 3 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43		ssdomg 8741	. . . . 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 9864	. . . . . . 7 ⊢ (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)))
46	1, 2, 45	mp2an 688	. . . . . 6 ⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴))
47		infdju 9895	. . . . . . 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 1449	. . . . . 6 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49		entr 8747	. . . . . 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 586	. . . . 5 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51		uncom 4083	. . . . 5 ⊢ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
52	50, 51	breqtrdi 5111	. . . 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 216	. 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 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558   class class class wbr 5070   × cxp 5578  dom cdm 5580  Oncon0 6251  ‘cfv 6418  ωcom 7687  1_oc1o 8260   ≈ cen 8688   ≼ cdom 8689   ⊔ cdju 9587  cardccrd 9624  ℵcale 9625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-dju 9590  df-card 9628  df-aleph 9629

This theorem is referenced by: (None)