MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephadd Structured version   Visualization version   GIF version

Theorem alephadd 9437
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
StepHypRef Expression
1 ovex 6718 . . . 4 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ∈ V
2 alephfnon 8926 . . . . . . . 8 ℵ Fn On
3 fndm 6028 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
42, 3ax-mp 5 . . . . . . 7 dom ℵ = On
54eleq2i 2722 . . . . . 6 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
65notbii 309 . . . . 5 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
74eleq2i 2722 . . . . . 6 (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
87notbii 309 . . . . 5 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
9 0ex 4823 . . . . . . . 8 ∅ ∈ V
10 cdaval 9030 . . . . . . . 8 ((∅ ∈ V ∧ ∅ ∈ V) → (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜})))
119, 9, 10mp2an 708 . . . . . . 7 (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
12 xpundi 5205 . . . . . . 7 (∅ × ({∅} ∪ {1𝑜})) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
13 0xp 5233 . . . . . . 7 (∅ × ({∅} ∪ {1𝑜})) = ∅
1411, 12, 133eqtr2i 2679 . . . . . 6 (∅ +𝑐 ∅) = ∅
15 ndmfv 6256 . . . . . . 7 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16 ndmfv 6256 . . . . . . 7 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
1715, 16oveqan12d 6709 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = (∅ +𝑐 ∅))
1815adantr 480 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
1916adantl 481 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
2018, 19uneq12d 3801 . . . . . . 7 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
21 un0 4000 . . . . . . 7 (∅ ∪ ∅) = ∅
2220, 21syl6eq 2701 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
2314, 17, 223eqtr4a 2711 . . . . 5 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
246, 8, 23syl2anbr 496 . . . 4 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
25 eqeng 8031 . . . 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))))
261, 24, 25mpsyl 68 . . 3 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
2726ex 449 . 2 𝐴 ∈ On → (¬ 𝐵 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
28 alephgeom 8943 . . 3 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
29 fvex 6239 . . . . 5 (ℵ‘𝐴) ∈ V
30 ssdomg 8043 . . . . 5 ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
3129, 30ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
32 alephon 8930 . . . . . 6 (ℵ‘𝐴) ∈ On
33 onenon 8813 . . . . . 6 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
3432, 33ax-mp 5 . . . . 5 (ℵ‘𝐴) ∈ dom card
35 alephon 8930 . . . . . 6 (ℵ‘𝐵) ∈ On
36 onenon 8813 . . . . . 6 ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
3735, 36ax-mp 5 . . . . 5 (ℵ‘𝐵) ∈ dom card
38 infcda 9068 . . . . 5 (((ℵ‘𝐴) ∈ dom card ∧ (ℵ‘𝐵) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
3934, 37, 38mp3an12 1454 . . . 4 (ω ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
4031, 39syl 17 . . 3 (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
4128, 40sylbi 207 . 2 (𝐴 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
42 alephgeom 8943 . . 3 (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43 fvex 6239 . . . . 5 (ℵ‘𝐵) ∈ V
44 ssdomg 8043 . . . . 5 ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)))
4543, 44ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))
46 cdacomen 9041 . . . . . 6 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴))
47 infcda 9068 . . . . . . 7 (((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐵)) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
4837, 34, 47mp3an12 1454 . . . . . 6 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49 entr 8049 . . . . . 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)))
5046, 48, 49sylancr 696 . . . . 5 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51 uncom 3790 . . . . 5 ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
5250, 51syl6breq 4726 . . . 4 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5345, 52syl 17 . . 3 (ω ⊆ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5442, 53sylbi 207 . 2 (𝐵 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5527, 41, 54pm2.61ii 177 1 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  wss 3607  c0 3948  {csn 4210   class class class wbr 4685   × cxp 5141  dom cdm 5143  Oncon0 5761   Fn wfn 5921  cfv 5926  (class class class)co 6690  ωcom 7107  1𝑜c1o 7598  cen 7994  cdom 7995  cardccrd 8799  cale 8800   +𝑐 ccda 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-card 8803  df-aleph 8804  df-cda 9028
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator