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Theorem cardalephex 10081
Description: Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.)
Assertion
Ref Expression
cardalephex (Ο‰ βŠ† 𝐴 β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯)))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem cardalephex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . 5 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ Ο‰ βŠ† 𝐴)
2 cardaleph 10080 . . . . . . 7 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ 𝐴 = (β„΅β€˜βˆ© {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)}))
32sseq2d 4006 . . . . . 6 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ (Ο‰ βŠ† 𝐴 ↔ Ο‰ βŠ† (β„΅β€˜βˆ© {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)})))
4 alephgeom 10073 . . . . . 6 (∩ {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)} ∈ On ↔ Ο‰ βŠ† (β„΅β€˜βˆ© {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)}))
53, 4bitr4di 289 . . . . 5 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ (Ο‰ βŠ† 𝐴 ↔ ∩ {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)} ∈ On))
61, 5mpbid 231 . . . 4 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ ∩ {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)} ∈ On)
7 fveq2 6881 . . . . 5 (π‘₯ = ∩ {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)} β†’ (β„΅β€˜π‘₯) = (β„΅β€˜βˆ© {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)}))
87rspceeqv 3625 . . . 4 ((∩ {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)} ∈ On ∧ 𝐴 = (β„΅β€˜βˆ© {𝑦 ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘¦)})) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
96, 2, 8syl2anc 583 . . 3 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
109ex 412 . 2 (Ο‰ βŠ† 𝐴 β†’ ((cardβ€˜π΄) = 𝐴 β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯)))
11 alephcard 10061 . . . 4 (cardβ€˜(β„΅β€˜π‘₯)) = (β„΅β€˜π‘₯)
12 fveq2 6881 . . . 4 (𝐴 = (β„΅β€˜π‘₯) β†’ (cardβ€˜π΄) = (cardβ€˜(β„΅β€˜π‘₯)))
13 id 22 . . . 4 (𝐴 = (β„΅β€˜π‘₯) β†’ 𝐴 = (β„΅β€˜π‘₯))
1411, 12, 133eqtr4a 2790 . . 3 (𝐴 = (β„΅β€˜π‘₯) β†’ (cardβ€˜π΄) = 𝐴)
1514rexlimivw 3143 . 2 (βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯) β†’ (cardβ€˜π΄) = 𝐴)
1610, 15impbid1 224 1 (Ο‰ βŠ† 𝐴 β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062  {crab 3424   βŠ† wss 3940  βˆ© cint 4940  Oncon0 6354  β€˜cfv 6533  Ο‰com 7848  cardccrd 9926  β„΅cale 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-oi 9501  df-har 9548  df-card 9930  df-aleph 9931
This theorem is referenced by:  infenaleph  10082  isinfcard  10083  alephfp  10099  alephval3  10101  dfac12k  10138  alephval2  10563  winalim2  10687  minregex  42774
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