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Theorem alephnbtwn 10068
Description: No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
alephnbtwn ((cardβ€˜π΅) = 𝐡 β†’ Β¬ ((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
Proof of Theorem alephnbtwn
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 alephon 10066 . . . . . . . 8 (β„΅β€˜π΄) ∈ On
2 id 22 . . . . . . . . . 10 ((cardβ€˜π΅) = 𝐡 β†’ (cardβ€˜π΅) = 𝐡)
3 cardon 9941 . . . . . . . . . 10 (cardβ€˜π΅) ∈ On
42, 3eqeltrrdi 2840 . . . . . . . . 9 ((cardβ€˜π΅) = 𝐡 β†’ 𝐡 ∈ On)
5 onenon 9946 . . . . . . . . 9 (𝐡 ∈ On β†’ 𝐡 ∈ dom card)
64, 5syl 17 . . . . . . . 8 ((cardβ€˜π΅) = 𝐡 β†’ 𝐡 ∈ dom card)
7 cardsdomel 9971 . . . . . . . 8 (((β„΅β€˜π΄) ∈ On ∧ 𝐡 ∈ dom card) β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 ↔ (β„΅β€˜π΄) ∈ (cardβ€˜π΅)))
81, 6, 7sylancr 585 . . . . . . 7 ((cardβ€˜π΅) = 𝐡 β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 ↔ (β„΅β€˜π΄) ∈ (cardβ€˜π΅)))
9 eleq2 2820 . . . . . . 7 ((cardβ€˜π΅) = 𝐡 β†’ ((β„΅β€˜π΄) ∈ (cardβ€˜π΅) ↔ (β„΅β€˜π΄) ∈ 𝐡))
108, 9bitrd 278 . . . . . 6 ((cardβ€˜π΅) = 𝐡 β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 ↔ (β„΅β€˜π΄) ∈ 𝐡))
1110adantl 480 . . . . 5 ((𝐴 ∈ On ∧ (cardβ€˜π΅) = 𝐡) β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 ↔ (β„΅β€˜π΄) ∈ 𝐡))
12 alephsuc 10065 . . . . . . . . . . 11 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = (harβ€˜(β„΅β€˜π΄)))
13 onenon 9946 . . . . . . . . . . . 12 ((β„΅β€˜π΄) ∈ On β†’ (β„΅β€˜π΄) ∈ dom card)
14 harval2 9994 . . . . . . . . . . . 12 ((β„΅β€˜π΄) ∈ dom card β†’ (harβ€˜(β„΅β€˜π΄)) = ∩ {π‘₯ ∈ On ∣ (β„΅β€˜π΄) β‰Ί π‘₯})
151, 13, 14mp2b 10 . . . . . . . . . . 11 (harβ€˜(β„΅β€˜π΄)) = ∩ {π‘₯ ∈ On ∣ (β„΅β€˜π΄) β‰Ί π‘₯}
1612, 15eqtrdi 2786 . . . . . . . . . 10 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = ∩ {π‘₯ ∈ On ∣ (β„΅β€˜π΄) β‰Ί π‘₯})
1716eleq2d 2817 . . . . . . . . 9 (𝐴 ∈ On β†’ (𝐡 ∈ (β„΅β€˜suc 𝐴) ↔ 𝐡 ∈ ∩ {π‘₯ ∈ On ∣ (β„΅β€˜π΄) β‰Ί π‘₯}))
1817biimpd 228 . . . . . . . 8 (𝐴 ∈ On β†’ (𝐡 ∈ (β„΅β€˜suc 𝐴) β†’ 𝐡 ∈ ∩ {π‘₯ ∈ On ∣ (β„΅β€˜π΄) β‰Ί π‘₯}))
19 breq2 5151 . . . . . . . . 9 (π‘₯ = 𝐡 β†’ ((β„΅β€˜π΄) β‰Ί π‘₯ ↔ (β„΅β€˜π΄) β‰Ί 𝐡))
2019onnminsb 7789 . . . . . . . 8 (𝐡 ∈ On β†’ (𝐡 ∈ ∩ {π‘₯ ∈ On ∣ (β„΅β€˜π΄) β‰Ί π‘₯} β†’ Β¬ (β„΅β€˜π΄) β‰Ί 𝐡))
2118, 20sylan9 506 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐡 ∈ (β„΅β€˜suc 𝐴) β†’ Β¬ (β„΅β€˜π΄) β‰Ί 𝐡))
2221con2d 134 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 β†’ Β¬ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
234, 22sylan2 591 . . . . 5 ((𝐴 ∈ On ∧ (cardβ€˜π΅) = 𝐡) β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 β†’ Β¬ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
2411, 23sylbird 259 . . . 4 ((𝐴 ∈ On ∧ (cardβ€˜π΅) = 𝐡) β†’ ((β„΅β€˜π΄) ∈ 𝐡 β†’ Β¬ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
25 imnan 398 . . . 4 (((β„΅β€˜π΄) ∈ 𝐡 β†’ Β¬ 𝐡 ∈ (β„΅β€˜suc 𝐴)) ↔ Β¬ ((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
2624, 25sylib 217 . . 3 ((𝐴 ∈ On ∧ (cardβ€˜π΅) = 𝐡) β†’ Β¬ ((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
2726ex 411 . 2 (𝐴 ∈ On β†’ ((cardβ€˜π΅) = 𝐡 β†’ Β¬ ((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴))))
28 n0i 4332 . . . . . . 7 (𝐡 ∈ (β„΅β€˜suc 𝐴) β†’ Β¬ (β„΅β€˜suc 𝐴) = βˆ…)
29 alephfnon 10062 . . . . . . . . . 10 β„΅ Fn On
3029fndmi 6652 . . . . . . . . 9 dom β„΅ = On
3130eleq2i 2823 . . . . . . . 8 (suc 𝐴 ∈ dom β„΅ ↔ suc 𝐴 ∈ On)
32 ndmfv 6925 . . . . . . . 8 (Β¬ suc 𝐴 ∈ dom β„΅ β†’ (β„΅β€˜suc 𝐴) = βˆ…)
3331, 32sylnbir 330 . . . . . . 7 (Β¬ suc 𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = βˆ…)
3428, 33nsyl2 141 . . . . . 6 (𝐡 ∈ (β„΅β€˜suc 𝐴) β†’ suc 𝐴 ∈ On)
35 onsucb 7807 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
3634, 35sylibr 233 . . . . 5 (𝐡 ∈ (β„΅β€˜suc 𝐴) β†’ 𝐴 ∈ On)
3736adantl 480 . . . 4 (((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴)) β†’ 𝐴 ∈ On)
3837con3i 154 . . 3 (Β¬ 𝐴 ∈ On β†’ Β¬ ((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
3938a1d 25 . 2 (Β¬ 𝐴 ∈ On β†’ ((cardβ€˜π΅) = 𝐡 β†’ Β¬ ((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴))))
4027, 39pm2.61i 182 1 ((cardβ€˜π΅) = 𝐡 β†’ Β¬ ((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {crab 3430  βˆ…c0 4321  βˆ© cint 4949   class class class wbr 5147  dom cdm 5675  Oncon0 6363  suc csuc 6365  β€˜cfv 6542   β‰Ί csdm 8940  harchar 9553  cardccrd 9932  β„΅cale 9933
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 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937
This theorem is referenced by:  alephnbtwn2  10069
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