Theorem winafpi 10589

Description: This theorem, which states that a nontrivial inaccessible cardinal is its own aleph number, is stated here in inference form, where the assumptions are in the hypotheses rather than an antecedent. Often, we use dedth 4531 to turn this type of statement into the closed form statement winafp 10588, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10588 using this theorem and dedth 4531, in ZFC. (You can prove this if you use ax-groth 10714, though.) (Contributed by Mario Carneiro, 28-May-2014.)

Hypotheses

Ref	Expression
winafp.1	⊢ 𝐴 ∈ Inaccw
winafp.2	⊢ 𝐴 ≠ ω

Assertion

Ref	Expression
winafpi	⊢ (ℵ‘𝐴) = 𝐴

Proof of Theorem winafpi

Step	Hyp	Ref	Expression
1		winafp.1	. 2 ⊢ 𝐴 ∈ Inaccw
2		winafp.2	. 2 ⊢ 𝐴 ≠ ω
3		winafp 10588	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ (ℵ‘𝐴) = 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ‘cfv 6481  ωcom 7796  ℵcale 9829  Inacc_wcwina 10573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-smo 8266  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-oi 9396  df-har 9443  df-card 9832  df-aleph 9833  df-cf 9834  df-acn 9835  df-wina 10575

This theorem is referenced by: (None)