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 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))

StepHypRef Expression
1 ovex 6577 . . . 4 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ∈ V
2 alephfnon 8771 . . . . . . . 8 ℵ Fn On
3 fndm 5904 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
42, 3ax-mp 5 . . . . . . 7 dom ℵ = On
54eleq2i 2680 . . . . . 6 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
65notbii 309 . . . . 5 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
74eleq2i 2680 . . . . . 6 (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
87notbii 309 . . . . 5 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
9 0ex 4718 . . . . . . . 8 ∅ ∈ V
10 cdaval 8875 . . . . . . . 8 ((∅ ∈ V ∧ ∅ ∈ V) → (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜})))
119, 9, 10mp2an 704 . . . . . . 7 (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
12 xpundi 5094 . . . . . . 7 (∅ × ({∅} ∪ {1𝑜})) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
13 0xp 5122 . . . . . . 7 (∅ × ({∅} ∪ {1𝑜})) = ∅
1411, 12, 133eqtr2i 2638 . . . . . 6 (∅ +𝑐 ∅) = ∅
15 ndmfv 6128 . . . . . . 7 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16 ndmfv 6128 . . . . . . 7 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
1715, 16oveqan12d 6568 . . . . . 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 3730 . . . . . . 7 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
21 un0 3919 . . . . . . 7 (∅ ∪ ∅) = ∅
2220, 21syl6eq 2660 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
2314, 17, 223eqtr4a 2670 . . . . 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 7875 . . . 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 66 . . 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 8788 . . 3 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
29 fvex 6113 . . . . 5 (ℵ‘𝐴) ∈ V
30 ssdomg 7887 . . . . 5 ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
3129, 30ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
32 alephon 8775 . . . . . 6 (ℵ‘𝐴) ∈ On
33 onenon 8658 . . . . . 6 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
3432, 33ax-mp 5 . . . . 5 (ℵ‘𝐴) ∈ dom card
35 alephon 8775 . . . . . 6 (ℵ‘𝐵) ∈ On
36 onenon 8658 . . . . . 6 ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
3735, 36ax-mp 5 . . . . 5 (ℵ‘𝐵) ∈ dom card
38 infcda 8913 . . . . 5 (((ℵ‘𝐴) ∈ dom card ∧ (ℵ‘𝐵) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
3934, 37, 38mp3an12 1406 . . . 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 206 . 2 (𝐴 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
42 alephgeom 8788 . . 3 (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43 fvex 6113 . . . . 5 (ℵ‘𝐵) ∈ V
44 ssdomg 7887 . . . . 5 ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)))
4543, 44ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))
46 cdacomen 8886 . . . . . 6 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴))
47 infcda 8913 . . . . . . 7 (((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐵)) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
4837, 34, 47mp3an12 1406 . . . . . 6 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49 entr 7894 . . . . . 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 694 . . . . 5 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51 uncom 3719 . . . . 5 ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
5250, 51syl6breq 4624 . . . 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 206 . 2 (𝐵 ∈ On → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
5527, 41, 54pm2.61ii 176 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 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  dom cdm 5038  Oncon0 5640   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440   ≈ cen 7838   ≼ cdom 7839  cardccrd 8644  ℵcale 8645   +𝑐 ccda 8872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649  df-cda 8873 This theorem is referenced by: (None)
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