Theorem winafpi 10621

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 4526 to turn this type of statement into the closed form statement winafp 10620, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10620 using this theorem and dedth 4526, in ZFC. (You can prove this if you use ax-groth 10746, 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 10620	. 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 6499  ωcom 7817  ℵcale 9860  Inacc_wcwina 10605

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-inf2 9562

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-smo 8286  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-oi 9425  df-har 9472  df-card 9863  df-aleph 9864  df-cf 9865  df-acn 9866  df-wina 10607

This theorem is referenced by: (None)