Theorem ttac 40561

Description: Tarski's theorem about choice: infxpidm 10176 is equivalent to ax-ac 10073. (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 9751	. 2 ⊢ (CHOICE ↔ dom card = V)
2		vex 3412	. . . . . 6 ⊢ 𝑐 ∈ V
3		eleq2 2826	. . . . . 6 ⊢ (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
4	2, 3	mpbiri 261	. . . . 5 ⊢ (dom card = V → 𝑐 ∈ dom card)
5		infxpidm2 9631	. . . . . 6 ⊢ ((𝑐 ∈ dom card ∧ ω ≼ 𝑐) → (𝑐 × 𝑐) ≈ 𝑐)
6	5	ex 416	. . . . 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 1935	. . 3 ⊢ (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9		finnum 9564	. . . . . . 7 ⊢ (𝑎 ∈ Fin → 𝑎 ∈ dom card)
10	9	adantl 485	. . . . . 6 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11		harcl 9175	. . . . . . . . 9 ⊢ (har‘𝑎) ∈ On
12		onenon 9565	. . . . . . . . 9 ⊢ ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (har‘𝑎) ∈ dom card
14		fvex 6730	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ∈ V
15		vex 3412	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
16	14, 15	unex 7531	. . . . . . . . . . . . 13 ⊢ ((har‘𝑎) ∪ 𝑎) ∈ V
17		harinf 40559	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
18	15, 17	mpan 690	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19		ssun1 4086	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
20	18, 19	sstrdi 3913	. . . . . . . . . . . . 13 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21		ssdomg 8674	. . . . . . . . . . . . 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 5057	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24		xpeq12 5576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = ((har‘𝑎) ∪ 𝑎) ∧ 𝑐 = ((har‘𝑎) ∪ 𝑎)) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
25	24	anidms 570	. . . . . . . . . . . . . . 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 5066	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((𝑐 × 𝑐) ≈ 𝑐 ↔ (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
28	23, 27	imbi12d 348	. . . . . . . . . . . . 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 3520	. . . . . . . . . . . 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 410	. . . . . . . . . 10 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))
32		harndom 9178	. . . . . . . . . . . 12 ⊢ ¬ (har‘𝑎) ≼ 𝑎
33		ssdomg 8674	. . . . . . . . . . . . . 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 8681	. . . . . . . . . . . . 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 200	. . . . . . . . . . 11 ⊢ ¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎
38		unxpwdom2 9204	. . . . . . . . . . 11 ⊢ ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎))
39		orel2 891	. . . . . . . . . . 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 9678	. . . . . . . . . 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 221	. . . . . . . 8 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎))
45		numdom 9652	. . . . . . . 8 ⊢ (((har‘𝑎) ∈ dom card ∧ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
46	13, 44, 45	sylancr 590	. . . . . . 7 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
47		ssun2 4087	. . . . . . 7 ⊢ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48		ssnum 9653	. . . . . . 7 ⊢ ((((har‘𝑎) ∪ 𝑎) ∈ dom card ∧ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)) → 𝑎 ∈ dom card)
49	46, 47, 48	sylancl 589	. . . . . 6 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
50	10, 49	pm2.61dan 813	. . . . 5 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → 𝑎 ∈ dom card)
51	50	alrimiv 1935	. . . 4 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52		eqv 3417	. . . 4 ⊢ (dom card = V ↔ ∀𝑎 𝑎 ∈ dom card)
53	51, 52	sylibr 237	. . 3 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → dom card = V)
54	8, 53	impbii 212	. 2 ⊢ (dom card = V ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
55	1, 54	bitri 278	1 ⊢ (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847  ∀wal 1541   = wceq 1543   ∈ wcel 2110  Vcvv 3408   ∪ cun 3864   ⊆ wss 3866   class class class wbr 5053   × cxp 5549  dom cdm 5551  Oncon0 6213  ‘cfv 6380  ωcom 7644   ≈ cen 8623   ≼ cdom 8624  Fincfn 8626  harchar 9172   ≼^* cwdom 9180  cardccrd 9551  CHOICEwac 9729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-oi 9126  df-har 9173  df-wdom 9181  df-card 9555  df-acn 9558  df-ac 9730

This theorem is referenced by: (None)