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Theorem alephadd 9591
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 6841 . . . 4 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ∈ V
2 alephfnon 9078 . . . . . . . 8 ℵ Fn On
3 fndm 6151 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
42, 3ax-mp 5 . . . . . . 7 dom ℵ = On
54eleq2i 2831 . . . . . 6 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
65notbii 309 . . . . 5 𝐴 ∈ dom ℵ ↔ ¬ 𝐴 ∈ On)
74eleq2i 2831 . . . . . 6 (𝐵 ∈ dom ℵ ↔ 𝐵 ∈ On)
87notbii 309 . . . . 5 𝐵 ∈ dom ℵ ↔ ¬ 𝐵 ∈ On)
9 0ex 4942 . . . . . . . 8 ∅ ∈ V
10 cdaval 9184 . . . . . . . 8 ((∅ ∈ V ∧ ∅ ∈ V) → (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜})))
119, 9, 10mp2an 710 . . . . . . 7 (∅ +𝑐 ∅) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
12 xpundi 5328 . . . . . . 7 (∅ × ({∅} ∪ {1𝑜})) = ((∅ × {∅}) ∪ (∅ × {1𝑜}))
13 0xp 5356 . . . . . . 7 (∅ × ({∅} ∪ {1𝑜})) = ∅
1411, 12, 133eqtr2i 2788 . . . . . 6 (∅ +𝑐 ∅) = ∅
15 ndmfv 6379 . . . . . . 7 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
16 ndmfv 6379 . . . . . . 7 𝐵 ∈ dom ℵ → (ℵ‘𝐵) = ∅)
1715, 16oveqan12d 6832 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = (∅ +𝑐 ∅))
1815adantr 472 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐴) = ∅)
1916adantl 473 . . . . . . . 8 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → (ℵ‘𝐵) = ∅)
2018, 19uneq12d 3911 . . . . . . 7 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = (∅ ∪ ∅))
21 un0 4110 . . . . . . 7 (∅ ∪ ∅) = ∅
2220, 21syl6eq 2810 . . . . . 6 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) = ∅)
2314, 17, 223eqtr4a 2820 . . . . 5 ((¬ 𝐴 ∈ dom ℵ ∧ ¬ 𝐵 ∈ dom ℵ) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
246, 8, 23syl2anbr 498 . . . 4 ((¬ 𝐴 ∈ On ∧ ¬ 𝐵 ∈ On) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
25 eqeng 8155 . . . 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 9095 . . 3 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
29 fvex 6362 . . . . 5 (ℵ‘𝐴) ∈ V
30 ssdomg 8167 . . . . 5 ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
3129, 30ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
32 alephon 9082 . . . . . 6 (ℵ‘𝐴) ∈ On
33 onenon 8965 . . . . . 6 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
3432, 33ax-mp 5 . . . . 5 (ℵ‘𝐴) ∈ dom card
35 alephon 9082 . . . . . 6 (ℵ‘𝐵) ∈ On
36 onenon 8965 . . . . . 6 ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card)
3735, 36ax-mp 5 . . . . 5 (ℵ‘𝐵) ∈ dom card
38 infcda 9222 . . . . 5 (((ℵ‘𝐴) ∈ dom card ∧ (ℵ‘𝐵) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
3934, 37, 38mp3an12 1563 . . . 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 9095 . . 3 (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵))
43 fvex 6362 . . . . 5 (ℵ‘𝐵) ∈ V
44 ssdomg 8167 . . . . 5 ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)))
4543, 44ax-mp 5 . . . 4 (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))
46 cdacomen 9195 . . . . . 6 ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴))
47 infcda 9222 . . . . . . 7 (((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐵)) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
4837, 34, 47mp3an12 1563 . . . . . 6 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐵) +𝑐 (ℵ‘𝐴)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
49 entr 8173 . . . . . 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 698 . . . . 5 (ω ≼ (ℵ‘𝐵) → ((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐵) ∪ (ℵ‘𝐴)))
51 uncom 3900 . . . . 5 ((ℵ‘𝐵) ∪ (ℵ‘𝐴)) = ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
5250, 51syl6breq 4845 . . . 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 1632  wcel 2139  Vcvv 3340  cun 3713  wss 3715  c0 4058  {csn 4321   class class class wbr 4804   × cxp 5264  dom cdm 5266  Oncon0 5884   Fn wfn 6044  cfv 6049  (class class class)co 6813  ωcom 7230  1𝑜c1o 7722  cen 8118  cdom 8119  cardccrd 8951  cale 8952   +𝑐 ccda 9181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-oi 8580  df-har 8628  df-card 8955  df-aleph 8956  df-cda 9182
This theorem is referenced by: (None)
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