Theorem alephiso3 44497

Description: ℵ 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.

Step	Hyp	Ref	Expression
1		alephiso2 44496	. 2 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
2		omelon 9625	. . . . . 6 ⊢ ω ∈ On
3		elrncard 44475	. . . . . . 7 ⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥))
4	3	simplbi 502	. . . . . 6 ⊢ (𝑥 ∈ ran card → 𝑥 ∈ On)
5		ontri1 6387	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
6	2, 4, 5	sylancr 599	. . . . 5 ⊢ (𝑥 ∈ ran card → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
7	6	rabbiia 3416	. . . 4 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω}
8		dfdif2 3908	. . . 4 ⊢ (ran card ∖ ω) = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω}
9	7, 8	eqtr4i 2786	. . 3 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω)
10		isoeq5 7318	. . 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))))
11	9, 10	ax-mp 5	. 2 ⊢ (ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) ↔ ℵ Isom E , ≺ (On, (ran card ∖ ω)))
12	1, 11	mpbi 233	1 ⊢ ℵ Isom E , ≺ (On, (ran card ∖ ω))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   ∖ cdif 3896   ⊆ wss 3899   class class class wbr 5103   E cep 5547  ran crn 5649  Oncon0 6352   Isom wiso 6529  ωcom 7861   ≈ cen 8949   ≺ csdm 8951  cardccrd 9973  ℵcale 9974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-isom 6537  df-riota 7366  df-ov 7412  df-om 7862  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8455  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-oi 9482  df-har 9529  df-card 9977  df-aleph 9978

This theorem is used by: (None)