Theorem axac3 10470

Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10469 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 10469	. . 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 1828	. 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 10173	. 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 234	1 ⊢ CHOICE

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568  ∃wex 1812  CHOICEwac 10122

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-ac2 10469

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ac 10123

This theorem is used by:  ackm  10471  axac  10473  axaci  10474  cardeqv  10475  fin71ac  10540  lbsex  21358  ptcls  23848  ptcmp  24290  kardcard  35702  axac10  43882