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Theorem alephnbtwn 10015
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 10013 . . . . . . . 8 (β„΅β€˜π΄) ∈ On
2 id 22 . . . . . . . . . 10 ((cardβ€˜π΅) = 𝐡 β†’ (cardβ€˜π΅) = 𝐡)
3 cardon 9888 . . . . . . . . . 10 (cardβ€˜π΅) ∈ On
42, 3eqeltrrdi 2843 . . . . . . . . 9 ((cardβ€˜π΅) = 𝐡 β†’ 𝐡 ∈ On)
5 onenon 9893 . . . . . . . . 9 (𝐡 ∈ On β†’ 𝐡 ∈ dom card)
64, 5syl 17 . . . . . . . 8 ((cardβ€˜π΅) = 𝐡 β†’ 𝐡 ∈ dom card)
7 cardsdomel 9918 . . . . . . . 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 10012 . . . . . . . . . . 11 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = (harβ€˜(β„΅β€˜π΄)))
13 onenon 9893 . . . . . . . . . . . 12 ((β„΅β€˜π΄) ∈ On β†’ (β„΅β€˜π΄) ∈ dom card)
14 harval2 9941 . . . . . . . . . . . 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 5113 . . . . . . . . 9 (π‘₯ = 𝐡 β†’ ((β„΅β€˜π΄) β‰Ί π‘₯ ↔ (β„΅β€˜π΄) β‰Ί 𝐡))
2019onnminsb 7738 . . . . . . . 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 4297 . . . . . . 7 (𝐡 ∈ (β„΅β€˜suc 𝐴) β†’ Β¬ (β„΅β€˜suc 𝐴) = βˆ…)
29 alephfnon 10009 . . . . . . . . . 10 β„΅ Fn On
3029fndmi 6610 . . . . . . . . 9 dom β„΅ = On
3130eleq2i 2826 . . . . . . . 8 (suc 𝐴 ∈ dom β„΅ ↔ suc 𝐴 ∈ On)
32 ndmfv 6881 . . . . . . . 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 7756 . . . . . 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 3406  βˆ…c0 4286  βˆ© cint 4911   class class class wbr 5109  dom cdm 5637  Oncon0 6321  suc csuc 6323  β€˜cfv 6500   β‰Ί csdm 8888  harchar 9500  cardccrd 9879  β„΅cale 9880
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-oi 9454  df-har 9501  df-card 9883  df-aleph 9884
This theorem is referenced by:  alephnbtwn2  10016
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