Theorem axac3 10377

Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10376 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 10376	. . 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 1797	. 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 10080	. 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 848  ∀wal 1540  ∃wex 1781  CHOICEwac 10028

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  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 10029

This theorem is referenced by:  ackm  10378  axac  10380  axaci  10381  cardeqv  10382  fin71ac  10446  lbsex  21155  ptcls  23591  ptcmp  24033  axac10  43479