Theorem ttac 44309

Description: Tarski's theorem about choice: infxpidm 10685 is equivalent to ax-ac 10576. (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 10255	. 2 ⊢ (CHOICE ↔ dom card = V)
2		vex 3458	. . . . . 6 ⊢ 𝑐 ∈ V
3		eleq2 2853	. . . . . 6 ⊢ (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
4	2, 3	mpbiri 261	. . . . 5 ⊢ (dom card = V → 𝑐 ∈ dom card)
5		infxpidm2 10135	. . . . . 6 ⊢ ((𝑐 ∈ dom card ∧ ω ≼ 𝑐) → (𝑐 × 𝑐) ≈ 𝑐)
6	5	ex 418	. . . . 5 ⊢ (𝑐 ∈ dom card → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
7	4, 6	syl 18	. . . 4 ⊢ (dom card = V → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
8	7	alrimiv 1962	. . 3 ⊢ (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9		finnum 10066	. . . . . . 7 ⊢ (𝑎 ∈ Fin → 𝑎 ∈ dom card)
10	9	adantl 487	. . . . . 6 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11		harcl 9590	. . . . . . . . 9 ⊢ (har‘𝑎) ∈ On
12		onenon 10067	. . . . . . . . 9 ⊢ ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (har‘𝑎) ∈ dom card
14		fvex 6929	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ∈ V
15		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
16	14, 15	unex 7793	. . . . . . . . . . . . 13 ⊢ ((har‘𝑎) ∪ 𝑎) ∈ V
17		harinf 44307	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
18	15, 17	mpan 703	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19		ssun1 4179	. . . . . . . . . . . . . 14 ⊢ (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
20	18, 19	sstrdi 3925	. . . . . . . . . . . . 13 ⊢ (¬ 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21		ssdomg 9065	. . . . . . . . . . . . 13 ⊢ (((har‘𝑎) ∪ 𝑎) ∈ V → (ω ⊆ ((har‘𝑎) ∪ 𝑎) → ω ≼ ((har‘𝑎) ∪ 𝑎)))
22	16, 20, 21	mpsyl 69	. . . . . . . . . . . 12 ⊢ (¬ 𝑎 ∈ Fin → ω ≼ ((har‘𝑎) ∪ 𝑎))
23		breq2 5112	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24		xpeq12 5683	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = ((har‘𝑎) ∪ 𝑎) ∧ 𝑐 = ((har‘𝑎) ∪ 𝑎)) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
25	24	anidms 577	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
26		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → 𝑐 = ((har‘𝑎) ∪ 𝑎))
27	25, 26	breq12d 5121	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((𝑐 × 𝑐) ≈ 𝑐 ↔ (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
28	23, 27	imbi12d 347	. . . . . . . . . . . . 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 3563	. . . . . . . . . . . 12 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → (ω ≼ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
30	22, 29	syl5 35	. . . . . . . . . . 11 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → (¬ 𝑎 ∈ Fin → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
31	30	imp 412	. . . . . . . . . 10 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))
32		harndom 9593	. . . . . . . . . . . 12 ⊢ ¬ (har‘𝑎) ≼ 𝑎
33		ssdomg 9065	. . . . . . . . . . . . . 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 9072	. . . . . . . . . . . . 13 ⊢ (((har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎) ∧ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → (har‘𝑎) ≼ 𝑎)
36	34, 35	mpan 703	. . . . . . . . . . . 12 ⊢ (((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → (har‘𝑎) ≼ 𝑎)
37	32, 36	mto 200	. . . . . . . . . . 11 ⊢ ¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎
38		unxpwdom2 9619	. . . . . . . . . . 11 ⊢ ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎))
39		orel2 904	. . . . . . . . . . 11 ⊢ (¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → ((((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎)))
40	37, 38, 39	mpsyl 69	. . . . . . . . . 10 ⊢ ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎))
41	31, 40	syl 18	. . . . . . . . 9 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎))
42		wdomnumr 10182	. . . . . . . . . 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 10156	. . . . . . . 8 ⊢ (((har‘𝑎) ∈ dom card ∧ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
46	13, 44, 45	sylancr 599	. . . . . . 7 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
47		ssun2 4180	. . . . . . 7 ⊢ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48		ssnum 10157	. . . . . . 7 ⊢ ((((har‘𝑎) ∪ 𝑎) ∈ dom card ∧ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)) → 𝑎 ∈ dom card)
49	46, 47, 48	sylancl 598	. . . . . 6 ⊢ ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
50	10, 49	pm2.61dan 825	. . . . 5 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → 𝑎 ∈ dom card)
51	50	alrimiv 1962	. . . 4 ⊢ (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52		eqv 3464	. . . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152   class class class wbr 5108   × cxp 5656  dom cdm 5658  Oncon0 6388  ‘cfv 6566  ωcom 7911   ≈ cen 9008   ≼ cdom 9009  Fincfn 9011  harchar 9587   ≼^* cwdom 9595  cardccrd 10053  CHOICEwac 10233

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-oi 9541  df-har 9588  df-wdom 9596  df-card 10057  df-acn 10060  df-ac 10234

This theorem is used by: (None)