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Theorem alephadd 9791
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
StepHypRef Expression
1 fvex 6506 . . . . 5 (ℵ‘𝐴) ∈ V
2 fvex 6506 . . . . 5 (ℵ‘𝐵) ∈ V
3 djuex 9125 . . . . 5 (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V)
41, 2, 3mp2an 679 . . . 4 ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V
5 alephfnon 9279 . . . . . . . 8 ℵ Fn On
6 fndm 6282 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
75, 6ax-mp 5 . . . . . . 7 dom ℵ = On
87eleq2i 2851 . . . . . 6 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
98notbii 312 . . . . 5 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
107eleq2i 2851 . . . . . 6 (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
1110notbii 312 . . . . 5 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
12 df-dju 9118 . . . . . . 7 (∅ ⊔ ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
13 xpundir 5465 . . . . . . 7 (({∅} ∪ {1o}) × ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
14 xp0 5849 . . . . . . 7 (({∅} ∪ {1o}) × ∅) = ∅
1512, 13, 143eqtr2i 2802 . . . . . 6 (∅ ⊔ ∅) = ∅
16 ndmfv 6523 . . . . . . 7 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
17 ndmfv 6523 . . . . . . 7 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
18 djueq12 9121 . . . . . . 7 (((ℵ‘𝐴) = ∅ ∧ (ℵ‘𝐵) = ∅) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
1916, 17, 18syl2an 586 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
2016adantr 473 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
2117adantl 474 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
2220, 21uneq12d 4023 . . . . . . 7 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
23 un0 4224 . . . . . . 7 (∅ ∪ ∅) = ∅
2422, 23syl6eq 2824 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
2515, 19, 243eqtr4a 2834 . . . . 5 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
269, 11, 25syl2anbr 589 . . . 4 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
27 eqeng 8334 . . . 4 (((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V → (((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
284, 26, 27mpsyl 68 . . 3 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
2928ex 405 . 2 𝐴 ∈ On → (¬ 𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
30 alephgeom 9296 . . 3 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
31 ssdomg 8346 . . . . 5 ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
321, 31ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
33 alephon 9283 . . . . . 6 (ℵ‘𝐴) ∈ On
34 onenon 9166 . . . . . 6 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
3533, 34ax-mp 5 . . . . 5 (ℵ‘𝐴) ∈ dom card
36 alephon 9283 . . . . . 6 (ℵ‘𝐵) ∈ On
37 onenon 9166 . . . . . 6 ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
3836, 37ax-mp 5 . . . . 5 (ℵ‘𝐵) ∈ dom card
39 infdju 9422 . . . . 5 (((ℵ‘𝐴) ∈ dom card ∧ (ℵ‘𝐵) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
4035, 38, 39mp3an12 1430 . . . 4 (ω ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
4132, 40syl 17 . . 3 (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
4230, 41sylbi 209 . 2 (𝐴 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
43 alephgeom 9296 . . 3 (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
44 ssdomg 8346 . . . . 5 ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)))
452, 44ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))
46 djucomen 9395 . . . . . . 7 (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)))
471, 2, 46mp2an 679 . . . . . 6 ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴))
48 infdju 9422 . . . . . . 7 (((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐵)) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
4938, 35, 48mp3an12 1430 . . . . . 6 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
50 entr 8352 . . . . . 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)))
5147, 49, 50sylancr 578 . . . . 5 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
52 uncom 4012 . . . . 5 ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
5351, 52syl6breq 4964 . . . 4 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5445, 53syl 17 . . 3 (ω ⊆ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5543, 54sylbi 209 . 2 (𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5629, 42, 55pm2.61ii 178 1 ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wcel 2050  Vcvv 3409  cun 3821  wss 3823  c0 4172  {csn 4435   class class class wbr 4923   × cxp 5399  dom cdm 5401  Oncon0 6023   Fn wfn 6177  cfv 6182  ωcom 7390  1oc1o 7892  cen 8297  cdom 8298  cdju 9115  cardccrd 9152  cale 9153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8892
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-se 5361  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7495  df-2nd 7496  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-2o 7900  df-oadd 7903  df-er 8083  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-oi 8763  df-har 8811  df-dju 9118  df-card 9156  df-aleph 9157
This theorem is referenced by: (None)
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