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Theorem ttac 37122
Description: Tarski's theorem about choice: infxpidm 9344 is equivalent to ax-ac 9241. (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.
StepHypRef Expression
1 dfac10 8919 . 2 (CHOICE ↔ dom card = V)
2 vex 3193 . . . . . 6 𝑐 ∈ V
3 eleq2 2687 . . . . . 6 (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
42, 3mpbiri 248 . . . . 5 (dom card = V → 𝑐 ∈ dom card)
5 infxpidm2 8800 . . . . . 6 ((𝑐 ∈ dom card ∧ ω ≼ 𝑐) → (𝑐 × 𝑐) ≈ 𝑐)
65ex 450 . . . . 5 (𝑐 ∈ dom card → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
74, 6syl 17 . . . 4 (dom card = V → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
87alrimiv 1852 . . 3 (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9 finnum 8734 . . . . . . 7 (𝑎 ∈ Fin → 𝑎 ∈ dom card)
109adantl 482 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11 harcl 8426 . . . . . . . . 9 (har‘𝑎) ∈ On
12 onenon 8735 . . . . . . . . 9 ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
1311, 12ax-mp 5 . . . . . . . 8 (har‘𝑎) ∈ dom card
14 fvex 6168 . . . . . . . . . . . . . 14 (har‘𝑎) ∈ V
15 vex 3193 . . . . . . . . . . . . . 14 𝑎 ∈ V
1614, 15unex 6921 . . . . . . . . . . . . 13 ((har‘𝑎) ∪ 𝑎) ∈ V
17 harinf 37120 . . . . . . . . . . . . . . 15 ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
1815, 17mpan 705 . . . . . . . . . . . . . 14 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19 ssun1 3760 . . . . . . . . . . . . . 14 (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
2018, 19syl6ss 3600 . . . . . . . . . . . . 13 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21 ssdomg 7961 . . . . . . . . . . . . 13 (((har‘𝑎) ∪ 𝑎) ∈ V → (ω ⊆ ((har‘𝑎) ∪ 𝑎) → ω ≼ ((har‘𝑎) ∪ 𝑎)))
2216, 20, 21mpsyl 68 . . . . . . . . . . . 12 𝑎 ∈ Fin → ω ≼ ((har‘𝑎) ∪ 𝑎))
23 breq2 4627 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24 xpeq12 5104 . . . . . . . . . . . . . . . 16 ((𝑐 = ((har‘𝑎) ∪ 𝑎) ∧ 𝑐 = ((har‘𝑎) ∪ 𝑎)) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
2524anidms 676 . . . . . . . . . . . . . . 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))
2725, 26breq12d 4636 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((𝑐 × 𝑐) ≈ 𝑐 ↔ (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
2823, 27imbi12d 334 . . . . . . . . . . . . 13 (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ↔ (ω ≼ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))))
2916, 28spcv 3289 . . . . . . . . . . . 12 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → (ω ≼ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
3022, 29syl5 34 . . . . . . . . . . 11 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → (¬ 𝑎 ∈ Fin → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
3130imp 445 . . . . . . . . . 10 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))
32 harndom 8429 . . . . . . . . . . . 12 ¬ (har‘𝑎) ≼ 𝑎
33 ssdomg 7961 . . . . . . . . . . . . . 14 (((har‘𝑎) ∪ 𝑎) ∈ V → ((har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎) → (har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎)))
3416, 19, 33mp2 9 . . . . . . . . . . . . 13 (har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎)
35 domtr 7969 . . . . . . . . . . . . 13 (((har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎) ∧ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → (har‘𝑎) ≼ 𝑎)
3634, 35mpan 705 . . . . . . . . . . . 12 (((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → (har‘𝑎) ≼ 𝑎)
3732, 36mto 188 . . . . . . . . . . 11 ¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎
38 unxpwdom2 8453 . . . . . . . . . . 11 ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎))
39 orel2 398 . . . . . . . . . . 11 (¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → ((((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎)))
4037, 38, 39mpsyl 68 . . . . . . . . . 10 ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎))
4131, 40syl 17 . . . . . . . . 9 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎))
42 wdomnumr 8847 . . . . . . . . . 10 ((har‘𝑎) ∈ dom card → (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ↔ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)))
4313, 42ax-mp 5 . . . . . . . . 9 (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ↔ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎))
4441, 43sylib 208 . . . . . . . 8 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎))
45 numdom 8821 . . . . . . . 8 (((har‘𝑎) ∈ dom card ∧ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
4613, 44, 45sylancr 694 . . . . . . 7 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
47 ssun2 3761 . . . . . . 7 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48 ssnum 8822 . . . . . . 7 ((((har‘𝑎) ∪ 𝑎) ∈ dom card ∧ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)) → 𝑎 ∈ dom card)
4946, 47, 48sylancl 693 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
5010, 49pm2.61dan 831 . . . . 5 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → 𝑎 ∈ dom card)
5150alrimiv 1852 . . . 4 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52 eqv 3195 . . . 4 (dom card = V ↔ ∀𝑎 𝑎 ∈ dom card)
5351, 52sylibr 224 . . 3 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → dom card = V)
548, 53impbii 199 . 2 (dom card = V ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
551, 54bitri 264 1 (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1987  Vcvv 3190  cun 3558  wss 3560   class class class wbr 4623   × cxp 5082  dom cdm 5084  Oncon0 5692  cfv 5857  ωcom 7027  cen 7912  cdom 7913  Fincfn 7915  harchar 8421  * cwdom 8422  cardccrd 8721  CHOICEwac 8898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-oi 8375  df-har 8423  df-wdom 8424  df-card 8725  df-acn 8728  df-ac 8899
This theorem is referenced by: (None)
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