Theorem axac3 10456

Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10455 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axac3	⊢ CHOICE

Proof of Theorem axac3

Dummy variables 𝑤 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-ac2 10455	. . 3 ⊢ ∃𝑦∀𝑧∃𝑤∀𝑣((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑤 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧 ∈ 𝑤))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑤 ∈ 𝑧 ∧ 𝑤 ∈ 𝑦) ∧ ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) → 𝑣 = 𝑤)))))
2	1	ax-gen 1789	. 2 ⊢ ∀𝑥∃𝑦∀𝑧∃𝑤∀𝑣((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑤 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧 ∈ 𝑤))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑤 ∈ 𝑧 ∧ 𝑤 ∈ 𝑦) ∧ ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) → 𝑣 = 𝑤)))))
3		dfackm 10158	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑦∀𝑧∃𝑤∀𝑣((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑤 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧 ∈ 𝑤))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑤 ∈ 𝑧 ∧ 𝑤 ∈ 𝑦) ∧ ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) → 𝑣 = 𝑤))))))
4	2, 3	mpbir 230	1 ⊢ CHOICE

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844  ∀wal 1531  ∃wex 1773  CHOICEwac 10107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-ac2 10455

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ac 10108

This theorem is referenced by:  ackm  10457  axac  10459  axaci  10460  cardeqv  10461  fin71ac  10525  lbsex  21012  ptcls  23464  ptcmp  23906  axac10  42324