Theorem winafpi 10692

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 4586 to turn this type of statement into the closed form statement winafp 10691, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10691 using this theorem and dedth 4586, in ZFC. (You can prove this if you use ax-groth 10817, 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 10691	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴)
4	1, 2, 3	mp2an 690	1 ⊢ (ℵ‘𝐴) = 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ‘cfv 6543  ωcom 7854  ℵcale 9930  Inacc_wcwina 10676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-smo 8345  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-oi 9504  df-har 9551  df-card 9933  df-aleph 9934  df-cf 9935  df-acn 9936  df-wina 10678

This theorem is referenced by: (None)