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Theorem alephiso3 43549
Description: is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
Assertion
Ref Expression
alephiso3 ℵ Isom E , ≺ (On, (ran card ∖ ω))

Proof of Theorem alephiso3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephiso2 43548 . 2 ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
2 omelon 9684 . . . . . 6 ω ∈ On
3 elrncard 43527 . . . . . . 7 (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 ¬ 𝑦𝑥))
43simplbi 497 . . . . . 6 (𝑥 ∈ ran card → 𝑥 ∈ On)
5 ontri1 6420 . . . . . 6 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
62, 4, 5sylancr 587 . . . . 5 (𝑥 ∈ ran card → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
76rabbiia 3437 . . . 4 {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω}
8 dfdif2 3972 . . . 4 (ran card ∖ ω) = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω}
97, 8eqtr4i 2766 . . 3 {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω)
10 isoeq5 7341 . . 3 ({𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω) → (ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) ↔ ℵ Isom E , ≺ (On, (ran card ∖ ω))))
119, 10ax-mp 5 . 2 (ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) ↔ ℵ Isom E , ≺ (On, (ran card ∖ ω)))
121, 11mpbi 230 1 ℵ Isom E , ≺ (On, (ran card ∖ ω))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  wss 3963   class class class wbr 5148   E cep 5588  ran crn 5690  Oncon0 6386   Isom wiso 6564  ωcom 7887  cen 8981  csdm 8983  cardccrd 9973  cale 9974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-har 9595  df-card 9977  df-aleph 9978
This theorem is referenced by: (None)
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