Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ttac Structured version   Visualization version   GIF version

Theorem ttac 37920
 Description: Tarski's theorem about choice: infxpidm 9422 is equivalent to ax-ac 9319. (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 8997 . 2 (CHOICE ↔ dom card = V)
2 vex 3234 . . . . . 6 𝑐 ∈ V
3 eleq2 2719 . . . . . 6 (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
42, 3mpbiri 248 . . . . 5 (dom card = V → 𝑐 ∈ dom card)
5 infxpidm2 8878 . . . . . 6 ((𝑐 ∈ dom card ∧ ω ≼ 𝑐) → (𝑐 × 𝑐) ≈ 𝑐)
65ex 449 . . . . 5 (𝑐 ∈ dom card → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
74, 6syl 17 . . . 4 (dom card = V → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
87alrimiv 1895 . . 3 (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9 finnum 8812 . . . . . . 7 (𝑎 ∈ Fin → 𝑎 ∈ dom card)
109adantl 481 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11 harcl 8507 . . . . . . . . 9 (har‘𝑎) ∈ On
12 onenon 8813 . . . . . . . . 9 ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
1311, 12ax-mp 5 . . . . . . . 8 (har‘𝑎) ∈ dom card
14 fvex 6239 . . . . . . . . . . . . . 14 (har‘𝑎) ∈ V
15 vex 3234 . . . . . . . . . . . . . 14 𝑎 ∈ V
1614, 15unex 6998 . . . . . . . . . . . . 13 ((har‘𝑎) ∪ 𝑎) ∈ V
17 harinf 37918 . . . . . . . . . . . . . . 15 ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
1815, 17mpan 706 . . . . . . . . . . . . . 14 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19 ssun1 3809 . . . . . . . . . . . . . 14 (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
2018, 19syl6ss 3648 . . . . . . . . . . . . 13 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21 ssdomg 8043 . . . . . . . . . . . . 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 4689 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24 xpeq12 5168 . . . . . . . . . . . . . . . 16 ((𝑐 = ((har‘𝑎) ∪ 𝑎) ∧ 𝑐 = ((har‘𝑎) ∪ 𝑎)) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
2524anidms 678 . . . . . . . . . . . . . . 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 4698 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((𝑐 × 𝑐) ≈ 𝑐 ↔ (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
2823, 27imbi12d 333 . . . . . . . . . . . . 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 3330 . . . . . . . . . . . 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 444 . . . . . . . . . 10 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))
32 harndom 8510 . . . . . . . . . . . 12 ¬ (har‘𝑎) ≼ 𝑎
33 ssdomg 8043 . . . . . . . . . . . . . 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 8050 . . . . . . . . . . . . 13 (((har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎) ∧ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → (har‘𝑎) ≼ 𝑎)
3634, 35mpan 706 . . . . . . . . . . . 12 (((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → (har‘𝑎) ≼ 𝑎)
3732, 36mto 188 . . . . . . . . . . 11 ¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎
38 unxpwdom2 8534 . . . . . . . . . . 11 ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎))
39 orel2 397 . . . . . . . . . . 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 8925 . . . . . . . . . 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 8899 . . . . . . . 8 (((har‘𝑎) ∈ dom card ∧ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
4613, 44, 45sylancr 696 . . . . . . 7 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
47 ssun2 3810 . . . . . . 7 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48 ssnum 8900 . . . . . . 7 ((((har‘𝑎) ∪ 𝑎) ∈ dom card ∧ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)) → 𝑎 ∈ dom card)
4946, 47, 48sylancl 695 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
5010, 49pm2.61dan 849 . . . . 5 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → 𝑎 ∈ dom card)
5150alrimiv 1895 . . . 4 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52 eqv 3236 . . . 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 382   ∧ wa 383  ∀wal 1521   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607   class class class wbr 4685   × cxp 5141  dom cdm 5143  Oncon0 5761  ‘cfv 5926  ωcom 7107   ≈ cen 7994   ≼ cdom 7995  Fincfn 7997  harchar 8502   ≼* cwdom 8503  cardccrd 8799  CHOICEwac 8976 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-wdom 8505  df-card 8803  df-acn 8806  df-ac 8977 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator