Theorem ttac 43027

Description: Tarski's theorem about choice: infxpidm 10581 is equivalent to ax-ac 10478. (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 10157	. 2 ⊢ (CHOICE ↔ dom card = V)
2		vex 3468	. . . . . 6 ⊢ 𝑐 ∈ V
3		eleq2 2824	. . . . . 6 ⊢ (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
4	2, 3	mpbiri 258	. . . . 5 ⊢ (dom card = V → 𝑐 ∈ dom card)
5		infxpidm2 10036	. . . . . 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 1927	. . 3 ⊢ (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9		finnum 9967	. . . . . . 7 ⊢ (𝑎 ∈ Fin → 𝑎 ∈ dom card)
10	9	adantl 481	. . . . . 6 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11		harcl 9578	. . . . . . . . 9 ⊢ (har‘𝑎) ∈ On
12		onenon 9968	. . . . . . . . 9 ⊢ ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (har‘𝑎) ∈ dom card
14		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ∈ V
15		vex 3468	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
16	14, 15	unex 7743	. . . . . . . . . . . . 13 ⊢ ((har‘𝑎) ∪ 𝑎) ∈ V
17		harinf 43025	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
18	15, 17	mpan 690	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19		ssun1 4158	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
20	18, 19	sstrdi 3976	. . . . . . . . . . . . 13 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21		ssdomg 9019	. . . . . . . . . . . . 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 5128	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24		xpeq12 5684	. . . . . . . . . . . . . . . 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 5137	. . . . . . . . . . . . . 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 3589	. . . . . . . . . . . 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 9581	. . . . . . . . . . . 12 ⊢ ¬ (har‘𝑎) ≼ 𝑎
33		ssdomg 9019	. . . . . . . . . . . . . 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 9026	. . . . . . . . . . . . 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 9607	. . . . . . . . . . 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 10083	. . . . . . . . . 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 10057	. . . . . . . 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 4159	. . . . . . 7 ⊢ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48		ssnum 10058	. . . . . . 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 1927	. . . 4 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52		eqv 3474	. . . 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 1538   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∪ cun 3929   ⊆ wss 3931   class class class wbr 5124   × cxp 5657  dom cdm 5659  Oncon0 6357  ‘cfv 6536  ωcom 7866   ≈ cen 8961   ≼ cdom 8962  Fincfn 8964  harchar 9575   ≼^* cwdom 9583  cardccrd 9954  CHOICEwac 10134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9529  df-har 9576  df-wdom 9584  df-card 9958  df-acn 9961  df-ac 10135

This theorem is referenced by: (None)