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Theorem alephadd 9251
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 6551 . . . 4 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ∈ V
2 alephfnon 8744 . . . . . . . 8 ℵ Fn On
3 fndm 5886 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
42, 3ax-mp 5 . . . . . . 7 dom ℵ = On
54eleq2i 2675 . . . . . 6 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
65notbii 308 . . . . 5 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
74eleq2i 2675 . . . . . 6 (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
87notbii 308 . . . . 5 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
9 0ex 4709 . . . . . . . 8 ∅ ∈ V
10 cdaval 8848 . . . . . . . 8 ((∅ ∈ V ∧ ∅ ∈ V) → (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜})))
119, 9, 10mp2an 703 . . . . . . 7 (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
12 xpundi 5080 . . . . . . 7 (∅ × ({∅} ∪ {1𝑜})) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
13 0xp 5108 . . . . . . 7 (∅ × ({∅} ∪ {1𝑜})) = ∅
1411, 12, 133eqtr2i 2633 . . . . . 6 (∅ +𝑐 ∅) = ∅
15 ndmfv 6109 . . . . . . 7 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16 ndmfv 6109 . . . . . . 7 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
1715, 16oveqan12d 6542 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = (∅ +𝑐 ∅))
1815adantr 479 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
1916adantl 480 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
2018, 19uneq12d 3725 . . . . . . 7 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
21 un0 3914 . . . . . . 7 (∅ ∪ ∅) = ∅
2220, 21syl6eq 2655 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
2314, 17, 223eqtr4a 2665 . . . . 5 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
246, 8, 23syl2anbr 495 . . . 4 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
25 eqeng 7848 . . . 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 65 . . 3 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
2726ex 448 . 2 𝐴 ∈ On → (¬ 𝐵 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))))
28 alephgeom 8761 . . 3 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
29 fvex 6094 . . . . 5 (ℵ‘𝐴) ∈ V
30 ssdomg 7860 . . . . 5 ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
3129, 30ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
32 alephon 8748 . . . . . 6 (ℵ‘𝐴) ∈ On
33 onenon 8631 . . . . . 6 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
3432, 33ax-mp 5 . . . . 5 (ℵ‘𝐴) ∈ dom card
35 alephon 8748 . . . . . 6 (ℵ‘𝐵) ∈ On
36 onenon 8631 . . . . . 6 ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
3735, 36ax-mp 5 . . . . 5 (ℵ‘𝐵) ∈ dom card
38 infcda 8886 . . . . 5 (((ℵ‘𝐴) ∈ dom card ∧ (ℵ‘𝐵) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
3934, 37, 38mp3an12 1405 . . . 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 205 . 2 (𝐴 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
42 alephgeom 8761 . . 3 (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43 fvex 6094 . . . . 5 (ℵ‘𝐵) ∈ V
44 ssdomg 7860 . . . . 5 ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)))
4543, 44ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))
46 cdacomen 8859 . . . . . 6 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴))
47 infcda 8886 . . . . . . 7 (((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐵)) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
4837, 34, 47mp3an12 1405 . . . . . 6 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49 entr 7867 . . . . . 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 693 . . . . 5 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51 uncom 3714 . . . . 5 ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
5250, 51syl6breq 4614 . . . 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 205 . 2 (𝐵 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5527, 41, 54pm2.61ii 175 1 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  cun 3533  wss 3535  c0 3869  {csn 4120   class class class wbr 4573   × cxp 5022  dom cdm 5024  Oncon0 5622   Fn wfn 5781  cfv 5786  (class class class)co 6523  ωcom 6930  1𝑜c1o 7413  cen 7811  cdom 7812  cardccrd 8617  cale 8618   +𝑐 ccda 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-oi 8271  df-har 8319  df-card 8621  df-aleph 8622  df-cda 8846
This theorem is referenced by: (None)
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