Theorem alephadd 9399

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) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))

Proof of Theorem alephadd

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 ⊢ ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ∈ V
2		alephfnon 8888	. . . . . . . 8 ⊢ ℵ Fn On
3		fndm 5990	. . . . . . . 8 ⊢ (ℵ Fn On → dom ℵ = On)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ dom ℵ = On
5	4	eleq2i 2693	. . . . . 6 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
6	5	notbii 310	. . . . 5 ⊢ (¬ 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
7	4	eleq2i 2693	. . . . . 6 ⊢ (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
8	7	notbii 310	. . . . 5 ⊢ (¬ 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
9		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
10		cdaval 8992	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜})))
11	9, 9, 10	mp2an 708	. . . . . . 7 ⊢ (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
12		xpundi 5171	. . . . . . 7 ⊢ (∅ × ({∅} ∪ {1𝑜})) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
13		0xp 5199	. . . . . . 7 ⊢ (∅ × ({∅} ∪ {1𝑜})) = ∅
14	11, 12, 13	3eqtr2i 2650	. . . . . 6 ⊢ (∅ +𝑐 ∅) = ∅
15		ndmfv 6218	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16		ndmfv 6218	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
17	15, 16	oveqan12d 6669	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = (∅ +𝑐 ∅))
18	15	adantr 481	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
19	16	adantl 482	. . . . . . . 8 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
20	18, 19	uneq12d 3768	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
21		un0 3967	. . . . . . 7 ⊢ (∅ ∪ ∅) = ∅
22	20, 21	syl6eq 2672	. . . . . 6 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
23	14, 17, 22	3eqtr4a 2682	. . . . 5 ⊢ ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
24	6, 8, 23	syl2anbr 497	. . . 4 ⊢ ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
25		eqeng 7989	. . . 4 ⊢ (((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ∈ V → (((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
26	1, 24, 25	mpsyl 68	. . 3 ⊢ ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
27	26	ex 450	. 2 ⊢ (¬ 𝐴 ∈ On → (¬ 𝐵 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
28		alephgeom 8905	. . 3 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
29		fvex 6201	. . . . 5 ⊢ (ℵ‘𝐴) ∈ V
30		ssdomg 8001	. . . . 5 ⊢ ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
31	29, 30	ax-mp 5	. . . 4 ⊢ (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
32		alephon 8892	. . . . . 6 ⊢ (ℵ‘𝐴) ∈ On
33		onenon 8775	. . . . . 6 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
34	32, 33	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐴) ∈ dom card
35		alephon 8892	. . . . . 6 ⊢ (ℵ‘𝐵) ∈ On
36		onenon 8775	. . . . . 6 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
37	35, 36	ax-mp 5	. . . . 5 ⊢ (ℵ‘𝐵) ∈ dom card
38		infcda 9030	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ dom card ∧ (ℵ‘𝐵) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
39	34, 37, 38	mp3an12 1414	. . . 4 ⊢ (ω ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
40	31, 39	syl 17	. . 3 ⊢ (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
41	28, 40	sylbi 207	. 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
42		alephgeom 8905	. . 3 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43		fvex 6201	. . . . 5 ⊢ (ℵ‘𝐵) ∈ V
44		ssdomg 8001	. . . . 5 ⊢ ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)))
45	43, 44	ax-mp 5	. . . 4 ⊢ (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))
46		cdacomen 9003	. . . . . 6 ⊢ ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴))
47		infcda 9030	. . . . . . 7 ⊢ (((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐵)) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
48	37, 34, 47	mp3an12 1414	. . . . . 6 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49		entr 8008	. . . . . 6 ⊢ ((((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ∧ ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴))) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
50	46, 48, 49	sylancr 695	. . . . 5 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51		uncom 3757	. . . . 5 ⊢ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
52	50, 51	syl6breq 4694	. . . 4 ⊢ (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
53	45, 52	syl 17	. . 3 ⊢ (ω ⊆ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
54	42, 53	sylbi 207	. 2 ⊢ (𝐵 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
55	27, 41, 54	pm2.61ii 177	1 ⊢ ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  dom cdm 5114  Oncon0 5723   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553   ≈ cen 7952   ≼ cdom 7953  cardccrd 8761  ℵcale 8762   +_𝑐 ccda 8989

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-se 5074  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-cda 8990

This theorem is referenced by: (None)