Theorem ttac 43288

Description: Tarski's theorem about choice: infxpidm 10472 is equivalent to ax-ac 10369. (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 10048	. 2 ⊢ (CHOICE ↔ dom card = V)
2		vex 3444	. . . . . 6 ⊢ 𝑐 ∈ V
3		eleq2 2825	. . . . . 6 ⊢ (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
4	2, 3	mpbiri 258	. . . . 5 ⊢ (dom card = V → 𝑐 ∈ dom card)
5		infxpidm2 9927	. . . . . 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 9860	. . . . . . 7 ⊢ (𝑎 ∈ Fin → 𝑎 ∈ dom card)
10	9	adantl 481	. . . . . 6 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11		harcl 9464	. . . . . . . . 9 ⊢ (har‘𝑎) ∈ On
12		onenon 9861	. . . . . . . . 9 ⊢ ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (har‘𝑎) ∈ dom card
14		fvex 6847	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ∈ V
15		vex 3444	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
16	14, 15	unex 7689	. . . . . . . . . . . . 13 ⊢ ((har‘𝑎) ∪ 𝑎) ∈ V
17		harinf 43286	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
18	15, 17	mpan 690	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19		ssun1 4130	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
20	18, 19	sstrdi 3946	. . . . . . . . . . . . 13 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21		ssdomg 8937	. . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24		xpeq12 5649	. . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . 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 3559	. . . . . . . . . . . 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 9467	. . . . . . . . . . . 12 ⊢ ¬ (har‘𝑎) ≼ 𝑎
33		ssdomg 8937	. . . . . . . . . . . . . 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 8944	. . . . . . . . . . . . 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 9493	. . . . . . . . . . 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 9974	. . . . . . . . . 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 9948	. . . . . . . 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 4131	. . . . . . 7 ⊢ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48		ssnum 9949	. . . . . . 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 3450	. . . 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 3440   ∪ cun 3899   ⊆ wss 3901   class class class wbr 5098   × cxp 5622  dom cdm 5624  Oncon0 6317  ‘cfv 6492  ωcom 7808   ≈ cen 8880   ≼ cdom 8881  Fincfn 8883  harchar 9461   ≼^* cwdom 9469  cardccrd 9847  CHOICEwac 10025

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9415  df-har 9462  df-wdom 9470  df-card 9851  df-acn 9854  df-ac 10026

This theorem is referenced by: (None)