Theorem ttac 43220

Description: Tarski's theorem about choice: infxpidm 10470 is equivalent to ax-ac 10367. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)

Assertion

Ref	Expression
ttac	⊢ (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))

Proof of Theorem ttac

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac10 10046	. 2 ⊢ (CHOICE ↔ dom card = V)
2		vex 3442	. . . . . 6 ⊢ 𝑐 ∈ V
3		eleq2 2823	. . . . . 6 ⊢ (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
4	2, 3	mpbiri 258	. . . . 5 ⊢ (dom card = V → 𝑐 ∈ dom card)
5		infxpidm2 9925	. . . . . 6 ⊢ ((𝑐 ∈ dom card ∧ ω ≼ 𝑐) → (𝑐 × 𝑐) ≈ 𝑐)
6	5	ex 412	. . . . 5 ⊢ (𝑐 ∈ dom card → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
7	4, 6	syl 17	. . . 4 ⊢ (dom card = V → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
8	7	alrimiv 1928	. . 3 ⊢ (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9		finnum 9858	. . . . . . 7 ⊢ (𝑎 ∈ Fin → 𝑎 ∈ dom card)
10	9	adantl 481	. . . . . 6 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11		harcl 9462	. . . . . . . . 9 ⊢ (har‘𝑎) ∈ On
12		onenon 9859	. . . . . . . . 9 ⊢ ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (har‘𝑎) ∈ dom card
14		fvex 6845	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ∈ V
15		vex 3442	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
16	14, 15	unex 7687	. . . . . . . . . . . . 13 ⊢ ((har‘𝑎) ∪ 𝑎) ∈ V
17		harinf 43218	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
18	15, 17	mpan 690	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19		ssun1 4128	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
20	18, 19	sstrdi 3944	. . . . . . . . . . . . 13 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21		ssdomg 8935	. . . . . . . . . . . . 13 ⊢ (((har‘𝑎) ∪ 𝑎) ∈ V → (ω ⊆ ((har‘𝑎) ∪ 𝑎) → ω ≼ ((har‘𝑎) ∪ 𝑎)))
22	16, 20, 21	mpsyl 68	. . . . . . . . . . . 12 ⊢ (¬ 𝑎 ∈ Fin → ω ≼ ((har‘𝑎) ∪ 𝑎))
23		breq2 5100	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24		xpeq12 5647	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = ((har‘𝑎) ∪ 𝑎) ∧ 𝑐 = ((har‘𝑎) ∪ 𝑎)) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
25	24	anidms 566	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
26		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → 𝑐 = ((har‘𝑎) ∪ 𝑎))
27	25, 26	breq12d 5109	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((𝑐 × 𝑐) ≈ 𝑐 ↔ (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
28	23, 27	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ↔ (ω ≼ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))))
29	16, 28	spcv 3557	. . . . . . . . . . . 12 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → (ω ≼ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
30	22, 29	syl5 34	. . . . . . . . . . 11 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → (¬ 𝑎 ∈ Fin → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
31	30	imp 406	. . . . . . . . . 10 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))
32		harndom 9465	. . . . . . . . . . . 12 ⊢ ¬ (har‘𝑎) ≼ 𝑎
33		ssdomg 8935	. . . . . . . . . . . . . 14 ⊢ (((har‘𝑎) ∪ 𝑎) ∈ V → ((har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎) → (har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎)))
34	16, 19, 33	mp2 9	. . . . . . . . . . . . 13 ⊢ (har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎)
35		domtr 8942	. . . . . . . . . . . . 13 ⊢ (((har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎) ∧ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → (har‘𝑎) ≼ 𝑎)
36	34, 35	mpan 690	. . . . . . . . . . . 12 ⊢ (((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → (har‘𝑎) ≼ 𝑎)
37	32, 36	mto 197	. . . . . . . . . . 11 ⊢ ¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎
38		unxpwdom2 9491	. . . . . . . . . . 11 ⊢ ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎))
39		orel2 890	. . . . . . . . . . 11 ⊢ (¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → ((((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎)))
40	37, 38, 39	mpsyl 68	. . . . . . . . . 10 ⊢ ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎))
41	31, 40	syl 17	. . . . . . . . 9 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎))
42		wdomnumr 9972	. . . . . . . . . 10 ⊢ ((har‘𝑎) ∈ dom card → (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ↔ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)))
43	13, 42	ax-mp 5	. . . . . . . . 9 ⊢ (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ↔ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎))
44	41, 43	sylib 218	. . . . . . . 8 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎))
45		numdom 9946	. . . . . . . 8 ⊢ (((har‘𝑎) ∈ dom card ∧ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
46	13, 44, 45	sylancr 587	. . . . . . 7 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
47		ssun2 4129	. . . . . . 7 ⊢ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48		ssnum 9947	. . . . . . 7 ⊢ ((((har‘𝑎) ∪ 𝑎) ∈ dom card ∧ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)) → 𝑎 ∈ dom card)
49	46, 47, 48	sylancl 586	. . . . . 6 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
50	10, 49	pm2.61dan 812	. . . . 5 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → 𝑎 ∈ dom card)
51	50	alrimiv 1928	. . . 4 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52		eqv 3448	. . . 4 ⊢ (dom card = V ↔ ∀𝑎 𝑎 ∈ dom card)
53	51, 52	sylibr 234	. . 3 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → dom card = V)
54	8, 53	impbii 209	. 2 ⊢ (dom card = V ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
55	1, 54	bitri 275	1 ⊢ (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∪ cun 3897   ⊆ wss 3899   class class class wbr 5096   × cxp 5620  dom cdm 5622  Oncon0 6315  ‘cfv 6490  ωcom 7806   ≈ cen 8878   ≼ cdom 8879  Fincfn 8881  harchar 9459   ≼^* cwdom 9467  cardccrd 9845  CHOICEwac 10023

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 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-oi 9413  df-har 9460  df-wdom 9468  df-card 9849  df-acn 9852  df-ac 10024

This theorem is referenced by: (None)