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Theorem alephnbtwn 10066
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 10064 . . . . . . . 8 (β„΅β€˜π΄) ∈ On
2 id 22 . . . . . . . . . 10 ((cardβ€˜π΅) = 𝐡 β†’ (cardβ€˜π΅) = 𝐡)
3 cardon 9939 . . . . . . . . . 10 (cardβ€˜π΅) ∈ On
42, 3eqeltrrdi 2843 . . . . . . . . 9 ((cardβ€˜π΅) = 𝐡 β†’ 𝐡 ∈ On)
5 onenon 9944 . . . . . . . . 9 (𝐡 ∈ On β†’ 𝐡 ∈ dom card)
64, 5syl 17 . . . . . . . 8 ((cardβ€˜π΅) = 𝐡 β†’ 𝐡 ∈ dom card)
7 cardsdomel 9969 . . . . . . . 8 (((β„΅β€˜π΄) ∈ On ∧ 𝐡 ∈ dom card) β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 ↔ (β„΅β€˜π΄) ∈ (cardβ€˜π΅)))
81, 6, 7sylancr 588 . . . . . . 7 ((cardβ€˜π΅) = 𝐡 β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 ↔ (β„΅β€˜π΄) ∈ (cardβ€˜π΅)))
9 eleq2 2823 . . . . . . 7 ((cardβ€˜π΅) = 𝐡 β†’ ((β„΅β€˜π΄) ∈ (cardβ€˜π΅) ↔ (β„΅β€˜π΄) ∈ 𝐡))
108, 9bitrd 279 . . . . . 6 ((cardβ€˜π΅) = 𝐡 β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 ↔ (β„΅β€˜π΄) ∈ 𝐡))
1110adantl 483 . . . . 5 ((𝐴 ∈ On ∧ (cardβ€˜π΅) = 𝐡) β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 ↔ (β„΅β€˜π΄) ∈ 𝐡))
12 alephsuc 10063 . . . . . . . . . . 11 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = (harβ€˜(β„΅β€˜π΄)))
13 onenon 9944 . . . . . . . . . . . 12 ((β„΅β€˜π΄) ∈ On β†’ (β„΅β€˜π΄) ∈ dom card)
14 harval2 9992 . . . . . . . . . . . 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 2789 . . . . . . . . . 10 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = ∩ {π‘₯ ∈ On ∣ (β„΅β€˜π΄) β‰Ί π‘₯})
1716eleq2d 2820 . . . . . . . . 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 5153 . . . . . . . . 9 (π‘₯ = 𝐡 β†’ ((β„΅β€˜π΄) β‰Ί π‘₯ ↔ (β„΅β€˜π΄) β‰Ί 𝐡))
2019onnminsb 7787 . . . . . . . 8 (𝐡 ∈ On β†’ (𝐡 ∈ ∩ {π‘₯ ∈ On ∣ (β„΅β€˜π΄) β‰Ί π‘₯} β†’ Β¬ (β„΅β€˜π΄) β‰Ί 𝐡))
2118, 20sylan9 509 . . . . . . 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 594 . . . . 5 ((𝐴 ∈ On ∧ (cardβ€˜π΅) = 𝐡) β†’ ((β„΅β€˜π΄) β‰Ί 𝐡 β†’ Β¬ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
2411, 23sylbird 260 . . . 4 ((𝐴 ∈ On ∧ (cardβ€˜π΅) = 𝐡) β†’ ((β„΅β€˜π΄) ∈ 𝐡 β†’ Β¬ 𝐡 ∈ (β„΅β€˜suc 𝐴)))
25 imnan 401 . . . 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 414 . 2 (𝐴 ∈ On β†’ ((cardβ€˜π΅) = 𝐡 β†’ Β¬ ((β„΅β€˜π΄) ∈ 𝐡 ∧ 𝐡 ∈ (β„΅β€˜suc 𝐴))))
28 n0i 4334 . . . . . . 7 (𝐡 ∈ (β„΅β€˜suc 𝐴) β†’ Β¬ (β„΅β€˜suc 𝐴) = βˆ…)
29 alephfnon 10060 . . . . . . . . . 10 β„΅ Fn On
3029fndmi 6654 . . . . . . . . 9 dom β„΅ = On
3130eleq2i 2826 . . . . . . . 8 (suc 𝐴 ∈ dom β„΅ ↔ suc 𝐴 ∈ On)
32 ndmfv 6927 . . . . . . . 8 (Β¬ suc 𝐴 ∈ dom β„΅ β†’ (β„΅β€˜suc 𝐴) = βˆ…)
3331, 32sylnbir 331 . . . . . . 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 7805 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
3634, 35sylibr 233 . . . . 5 (𝐡 ∈ (β„΅β€˜suc 𝐴) β†’ 𝐴 ∈ On)
3736adantl 483 . . . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433  βˆ…c0 4323  βˆ© cint 4951   class class class wbr 5149  dom cdm 5677  Oncon0 6365  suc csuc 6367  β€˜cfv 6544   β‰Ί csdm 8938  harchar 9551  cardccrd 9930  β„΅cale 9931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-oi 9505  df-har 9552  df-card 9934  df-aleph 9935
This theorem is referenced by:  alephnbtwn2  10067
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