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Theorem alephadd 10379
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 6817 . . . . 5 (ℵ‘𝐴) ∈ V
2 fvex 6817 . . . . 5 (ℵ‘𝐵) ∈ V
3 djuex 9710 . . . . 5 (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V)
41, 2, 3mp2an 690 . . . 4 ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V
5 alephfnon 9867 . . . . . . . 8 ℵ Fn On
65fndmi 6568 . . . . . . 7 dom ℵ = On
76eleq2i 2828 . . . . . 6 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
87notbii 320 . . . . 5 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
96eleq2i 2828 . . . . . 6 (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
109notbii 320 . . . . 5 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
11 df-dju 9703 . . . . . . 7 (∅ ⊔ ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
12 xpundir 5667 . . . . . . 7 (({∅} ∪ {1o}) × ∅) = (({∅} × ∅) ∪ ({1o} × ∅))
13 xp0 6076 . . . . . . 7 (({∅} ∪ {1o}) × ∅) = ∅
1411, 12, 133eqtr2i 2770 . . . . . 6 (∅ ⊔ ∅) = ∅
15 ndmfv 6836 . . . . . . 7 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16 ndmfv 6836 . . . . . . 7 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
17 djueq12 9706 . . . . . . 7 (((ℵ‘𝐴) = ∅ ∧ (ℵ‘𝐵) = ∅) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
1815, 16, 17syl2an 597 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = (∅ ⊔ ∅))
1915adantr 482 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
2016adantl 483 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
2119, 20uneq12d 4104 . . . . . . 7 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
22 un0 4330 . . . . . . 7 (∅ ∪ ∅) = ∅
2321, 22eqtrdi 2792 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
2414, 18, 233eqtr4a 2802 . . . . 5 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
258, 10, 24syl2anbr 600 . . . 4 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
26 eqeng 8807 . . . 4 (((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ∈ V → (((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
274, 25, 26mpsyl 68 . . 3 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
2827ex 414 . 2 𝐴 ∈ On → (¬ 𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
29 alephgeom 9884 . . 3 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
30 ssdomg 8821 . . . . 5 ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
311, 30ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
32 alephon 9871 . . . . . 6 (ℵ‘𝐴) ∈ On
33 onenon 9751 . . . . . 6 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
3432, 33ax-mp 5 . . . . 5 (ℵ‘𝐴) ∈ dom card
35 alephon 9871 . . . . . 6 (ℵ‘𝐵) ∈ On
36 onenon 9751 . . . . . 6 ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
3735, 36ax-mp 5 . . . . 5 (ℵ‘𝐵) ∈ dom card
38 infdju 10010 . . . . 5 (((ℵ‘𝐴) ∈ dom card ∧ (ℵ‘𝐵) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
3934, 37, 38mp3an12 1451 . . . 4 (ω ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
4031, 39syl 17 . . 3 (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
4129, 40sylbi 216 . 2 (𝐴 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
42 alephgeom 9884 . . 3 (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43 ssdomg 8821 . . . . 5 ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)))
442, 43ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))
45 djucomen 9979 . . . . . . 7 (((ℵ‘𝐴) ∈ V ∧ (ℵ‘𝐵) ∈ V) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)))
461, 2, 45mp2an 690 . . . . . 6 ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ⊔ (ℵ‘𝐴))
47 infdju 10010 . . . . . . 7 (((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐵)) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
4837, 34, 47mp3an12 1451 . . . . . 6 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) ⊔ (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49 entr 8827 . . . . . 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)))
5046, 48, 49sylancr 588 . . . . 5 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51 uncom 4093 . . . . 5 ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
5250, 51breqtrdi 5122 . . . 4 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5344, 52syl 17 . . 3 (ω ⊆ (ℵ‘𝐵) → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5442, 53sylbi 216 . 2 (𝐵 ∈ On → ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5528, 41, 54pm2.61ii 183 1 ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1539  wcel 2104  Vcvv 3437  cun 3890  wss 3892  c0 4262  {csn 4565   class class class wbr 5081   × cxp 5598  dom cdm 5600  Oncon0 6281  cfv 6458  ωcom 7744  1oc1o 8321  cen 8761  cdom 8762  cdju 9700  cardccrd 9737  cale 9738
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-inf2 9443
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-se 5556  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-2o 8329  df-oadd 8332  df-er 8529  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-oi 9313  df-har 9360  df-dju 9703  df-card 9741  df-aleph 9742
This theorem is referenced by: (None)
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