Theorem winafpi 10619

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 4520 to turn this type of statement into the closed form statement winafp 10618, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10618 using this theorem and dedth 4520, in ZFC. (You can prove this if you use ax-groth 10744, 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 10618	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴)
4	1, 2, 3	mp2an 698	1 ⊢ (ℵ‘𝐴) = 𝐴

Syntax hints:   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ‘cfv 6492  ωcom 7813  ℵcale 9858  Inacc_wcwina 10603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-smo 8283  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9422  df-har 9469  df-card 9861  df-aleph 9862  df-cf 9863  df-acn 9864  df-wina 10605

This theorem is referenced by: (None)