Theorem winafpi 10623

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 4540 to turn this type of statement into the closed form statement winafp 10622, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10622 using this theorem and dedth 4540, in ZFC. (You can prove this if you use ax-groth 10748, 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 10622	. 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴)
4	1, 2, 3	mp2an 693	1 ⊢ (ℵ‘𝐴) = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6502  ωcom 7820  ℵcale 9862  Inacc_wcwina 10607

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-smo 8290  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-map 8779  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-oi 9429  df-har 9476  df-card 9865  df-aleph 9866  df-cf 9867  df-acn 9868  df-wina 10609

This theorem is referenced by: (None)