Theorem axac3 10374

Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10373 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 10373	. . 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 1796	. 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 10077	. 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 231	1 ⊢ CHOICE

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1539  ∃wex 1780  CHOICEwac 10025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-ac2 10373

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ac 10026

This theorem is referenced by:  ackm  10375  axac  10377  axaci  10378  cardeqv  10379  fin71ac  10443  lbsex  21120  ptcls  23560  ptcmp  24002  axac10  43271