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Theorem ttac 43053
Description: Tarski's theorem about choice: infxpidm 10603 is equivalent to ax-ac 10500. (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 10179 . 2 (CHOICE ↔ dom card = V)
2 vex 3483 . . . . . 6 𝑐 ∈ V
3 eleq2 2829 . . . . . 6 (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
42, 3mpbiri 258 . . . . 5 (dom card = V → 𝑐 ∈ dom card)
5 infxpidm2 10058 . . . . . 6 ((𝑐 ∈ dom card ∧ ω ≼ 𝑐) → (𝑐 × 𝑐) ≈ 𝑐)
65ex 412 . . . . 5 (𝑐 ∈ dom card → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
74, 6syl 17 . . . 4 (dom card = V → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
87alrimiv 1926 . . 3 (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9 finnum 9989 . . . . . . 7 (𝑎 ∈ Fin → 𝑎 ∈ dom card)
109adantl 481 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11 harcl 9600 . . . . . . . . 9 (har‘𝑎) ∈ On
12 onenon 9990 . . . . . . . . 9 ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
1311, 12ax-mp 5 . . . . . . . 8 (har‘𝑎) ∈ dom card
14 fvex 6918 . . . . . . . . . . . . . 14 (har‘𝑎) ∈ V
15 vex 3483 . . . . . . . . . . . . . 14 𝑎 ∈ V
1614, 15unex 7765 . . . . . . . . . . . . 13 ((har‘𝑎) ∪ 𝑎) ∈ V
17 harinf 43051 . . . . . . . . . . . . . . 15 ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
1815, 17mpan 690 . . . . . . . . . . . . . 14 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19 ssun1 4177 . . . . . . . . . . . . . 14 (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
2018, 19sstrdi 3995 . . . . . . . . . . . . 13 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21 ssdomg 9041 . . . . . . . . . . . . 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 5146 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24 xpeq12 5709 . . . . . . . . . . . . . . . 16 ((𝑐 = ((har‘𝑎) ∪ 𝑎) ∧ 𝑐 = ((har‘𝑎) ∪ 𝑎)) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
2524anidms 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))
2725, 26breq12d 5155 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((𝑐 × 𝑐) ≈ 𝑐 ↔ (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
2823, 27imbi12d 344 . . . . . . . . . . . . 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 3604 . . . . . . . . . . . 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 406 . . . . . . . . . 10 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))
32 harndom 9603 . . . . . . . . . . . 12 ¬ (har‘𝑎) ≼ 𝑎
33 ssdomg 9041 . . . . . . . . . . . . . 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 9048 . . . . . . . . . . . . 13 (((har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎) ∧ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → (har‘𝑎) ≼ 𝑎)
3634, 35mpan 690 . . . . . . . . . . . 12 (((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → (har‘𝑎) ≼ 𝑎)
3732, 36mto 197 . . . . . . . . . . 11 ¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎
38 unxpwdom2 9629 . . . . . . . . . . 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)))
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 10105 . . . . . . . . . 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 218 . . . . . . . 8 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎))
45 numdom 10079 . . . . . . . 8 (((har‘𝑎) ∈ dom card ∧ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
4613, 44, 45sylancr 587 . . . . . . 7 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
47 ssun2 4178 . . . . . . 7 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48 ssnum 10080 . . . . . . 7 ((((har‘𝑎) ∪ 𝑎) ∈ dom card ∧ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)) → 𝑎 ∈ dom card)
4946, 47, 48sylancl 586 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
5010, 49pm2.61dan 812 . . . . 5 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → 𝑎 ∈ dom card)
5150alrimiv 1926 . . . 4 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52 eqv 3489 . . . 4 (dom card = V ↔ ∀𝑎 𝑎 ∈ dom card)
5351, 52sylibr 234 . . 3 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → dom card = V)
548, 53impbii 209 . 2 (dom card = V ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
551, 54bitri 275 1 (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  wal 1537   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  wss 3950   class class class wbr 5142   × cxp 5682  dom cdm 5684  Oncon0 6383  cfv 6560  ωcom 7888  cen 8983  cdom 8984  Fincfn 8986  harchar 9597  * cwdom 9605  cardccrd 9976  CHOICEwac 10156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-oi 9551  df-har 9598  df-wdom 9606  df-card 9980  df-acn 9983  df-ac 10157
This theorem is referenced by: (None)
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