Theorem axac3 9880

Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 9879 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 9879	. . 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 1792	. 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 9586	. 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 233	1 ⊢ CHOICE

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843  ∀wal 1531  ∃wex 1776  CHOICEwac 9535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-ac2 9879

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ac 9536

This theorem is referenced by:  ackm  9881  axac  9883  axaci  9884  cardeqv  9885  fin71ac  9949  lbsex  19931  ptcls  22218  ptcmp  22660  axac10  39623