Theorem ac5b 3574

Description: Equivalent of Axiom of Choice.

Hypothesis

Ref	Expression
ac5b.1	⊢ A ∈ V

Assertion

Ref	Expression
ac5b	⊢ (∀x ∈ A ¬ x = ∅ → ∃f(f:A–→∪A ∧ ∀x ∈ A (f ‘x) ∈ x))

Distinct variable group(s):   x,f,A

Proof of Theorem ac5b

Step	Hyp	Ref	Expression
1		ac5b.1	. . 3 ⊢ A ∈ V
2	1	ac5 3573	. 2 ⊢ ∃f(f Fn A ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))
3		19.42v 966	. . 3 ⊢ (∃f(∀x ∈ A ¬ x = ∅ ∧ (f Fn A ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))) ↔ (∀x ∈ A ¬ x = ∅ ∧ ∃f(f Fn A ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))))
4		chfnrn 2885	. . . . . . . . . 10 ⊢ ((f Fn A ∧ ∀x ∈ A (f ‘x) ∈ x) → ran f ⊆ ∪A)
5	4	exp 291	. . . . . . . . 9 ⊢ (f Fn A → (∀x ∈ A (f ‘x) ∈ x → ran f ⊆ ∪A))
6	5	anc2li 250	. . . . . . . 8 ⊢ (f Fn A → (∀x ∈ A (f ‘x) ∈ x → (f Fn A ∧ ran f ⊆ ∪A)))
7		df-f 2434	. . . . . . . 8 ⊢ (f:A–→∪A ↔ (f Fn A ∧ ran f ⊆ ∪A))
8	6, 7	syl6ibr 186	. . . . . . 7 ⊢ (f Fn A → (∀x ∈ A (f ‘x) ∈ x → f:A–→∪A))
9	8	impac 304	. . . . . 6 ⊢ ((f Fn A ∧ ∀x ∈ A (f ‘x) ∈ x) → (f:A–→∪A ∧ ∀x ∈ A (f ‘x) ∈ x))
10		r19.26 1289	. . . . . . 7 ⊢ (∀x ∈ A (¬ x = ∅ ∧ (¬ x = ∅ → (f ‘x) ∈ x)) ↔ (∀x ∈ A ¬ x = ∅ ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x)))
11		pm3.35 278	. . . . . . . 8 ⊢ ((¬ x = ∅ ∧ (¬ x = ∅ → (f ‘x) ∈ x)) → (f ‘x) ∈ x)
12	11	r19.20si 1254	. . . . . . 7 ⊢ (∀x ∈ A (¬ x = ∅ ∧ (¬ x = ∅ → (f ‘x) ∈ x)) → ∀x ∈ A (f ‘x) ∈ x)
13	10, 12	sylbir 176	. . . . . 6 ⊢ ((∀x ∈ A ¬ x = ∅ ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x)) → ∀x ∈ A (f ‘x) ∈ x)
14	9, 13	sylan2 346	. . . . 5 ⊢ ((f Fn A ∧ (∀x ∈ A ¬ x = ∅ ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))) → (f:A–→∪A ∧ ∀x ∈ A (f ‘x) ∈ x))
15	14	an1s 372	. . . 4 ⊢ ((∀x ∈ A ¬ x = ∅ ∧ (f Fn A ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))) → (f:A–→∪A ∧ ∀x ∈ A (f ‘x) ∈ x))
16	15	19.22i 723	. . 3 ⊢ (∃f(∀x ∈ A ¬ x = ∅ ∧ (f Fn A ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))) → ∃f(f:A–→∪A ∧ ∀x ∈ A (f ‘x) ∈ x))
17	3, 16	sylbir 176	. 2 ⊢ ((∀x ∈ A ¬ x = ∅ ∧ ∃f(f Fn A ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))) → ∃f(f:A–→∪A ∧ ∀x ∈ A (f ‘x) ∈ x))
18	2, 17	mpan2 519	1 ⊢ (∀x ∈ A ¬ x = ∅ → ∃f(f:A–→∪A ∧ ∀x ∈ A (f ‘x) ∈ x))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919  ran crn 2411   Fn wfn 2417  –→wf 2418   ‘cfv 2422

This theorem is referenced by:  ac6lem 3575

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-fr 2169  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438

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