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Theorem ttac 39035
Description: Tarski's theorem about choice: infxpidm 9782 is equivalent to ax-ac 9679. (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 9357 . 2 (CHOICE ↔ dom card = V)
2 vex 3418 . . . . . 6 𝑐 ∈ V
3 eleq2 2854 . . . . . 6 (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
42, 3mpbiri 250 . . . . 5 (dom card = V → 𝑐 ∈ dom card)
5 infxpidm2 9237 . . . . . 6 ((𝑐 ∈ dom card ∧ ω ≼ 𝑐) → (𝑐 × 𝑐) ≈ 𝑐)
65ex 405 . . . . 5 (𝑐 ∈ dom card → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
74, 6syl 17 . . . 4 (dom card = V → (ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
87alrimiv 1886 . . 3 (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9 finnum 9171 . . . . . . 7 (𝑎 ∈ Fin → 𝑎 ∈ dom card)
109adantl 474 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11 harcl 8820 . . . . . . . . 9 (har‘𝑎) ∈ On
12 onenon 9172 . . . . . . . . 9 ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
1311, 12ax-mp 5 . . . . . . . 8 (har‘𝑎) ∈ dom card
14 fvex 6512 . . . . . . . . . . . . . 14 (har‘𝑎) ∈ V
15 vex 3418 . . . . . . . . . . . . . 14 𝑎 ∈ V
1614, 15unex 7286 . . . . . . . . . . . . 13 ((har‘𝑎) ∪ 𝑎) ∈ V
17 harinf 39033 . . . . . . . . . . . . . . 15 ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
1815, 17mpan 677 . . . . . . . . . . . . . 14 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19 ssun1 4037 . . . . . . . . . . . . . 14 (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
2018, 19syl6ss 3870 . . . . . . . . . . . . 13 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21 ssdomg 8352 . . . . . . . . . . . . 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 4933 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24 xpeq12 5432 . . . . . . . . . . . . . . . 16 ((𝑐 = ((har‘𝑎) ∪ 𝑎) ∧ 𝑐 = ((har‘𝑎) ∪ 𝑎)) → (𝑐 × 𝑐) = (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)))
2524anidms 559 . . . . . . . . . . . . . . 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 4942 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → ((𝑐 × 𝑐) ≈ 𝑐 ↔ (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎)))
2823, 27imbi12d 337 . . . . . . . . . . . . 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 3524 . . . . . . . . . . . 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 398 . . . . . . . . . 10 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → (((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎))
32 harndom 8823 . . . . . . . . . . . 12 ¬ (har‘𝑎) ≼ 𝑎
33 ssdomg 8352 . . . . . . . . . . . . . 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 8359 . . . . . . . . . . . . 13 (((har‘𝑎) ≼ ((har‘𝑎) ∪ 𝑎) ∧ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎) → (har‘𝑎) ≼ 𝑎)
3634, 35mpan 677 . . . . . . . . . . . 12 (((har‘𝑎) ∪ 𝑎) ≼ 𝑎 → (har‘𝑎) ≼ 𝑎)
3732, 36mto 189 . . . . . . . . . . 11 ¬ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎
38 unxpwdom2 8847 . . . . . . . . . . 11 ((((har‘𝑎) ∪ 𝑎) × ((har‘𝑎) ∪ 𝑎)) ≈ ((har‘𝑎) ∪ 𝑎) → (((har‘𝑎) ∪ 𝑎) ≼* (har‘𝑎) ∨ ((har‘𝑎) ∪ 𝑎) ≼ 𝑎))
39 orel2 874 . . . . . . . . . . 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 9284 . . . . . . . . . 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 210 . . . . . . . 8 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎))
45 numdom 9258 . . . . . . . 8 (((har‘𝑎) ∈ dom card ∧ ((har‘𝑎) ∪ 𝑎) ≼ (har‘𝑎)) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
4613, 44, 45sylancr 578 . . . . . . 7 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → ((har‘𝑎) ∪ 𝑎) ∈ dom card)
47 ssun2 4038 . . . . . . 7 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48 ssnum 9259 . . . . . . 7 ((((har‘𝑎) ∪ 𝑎) ∈ dom card ∧ 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)) → 𝑎 ∈ dom card)
4946, 47, 48sylancl 577 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ ¬ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
5010, 49pm2.61dan 800 . . . . 5 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → 𝑎 ∈ dom card)
5150alrimiv 1886 . . . 4 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52 eqv 3423 . . . 4 (dom card = V ↔ ∀𝑎 𝑎 ∈ dom card)
5351, 52sylibr 226 . . 3 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → dom card = V)
548, 53impbii 201 . 2 (dom card = V ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
551, 54bitri 267 1 (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833  wal 1505   = wceq 1507  wcel 2050  Vcvv 3415  cun 3827  wss 3829   class class class wbr 4929   × cxp 5405  dom cdm 5407  Oncon0 6029  cfv 6188  ωcom 7396  cen 8303  cdom 8304  Fincfn 8306  harchar 8815  * cwdom 8816  cardccrd 9158  CHOICEwac 9335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-oi 8769  df-har 8817  df-wdom 8818  df-card 9162  df-acn 9165  df-ac 9336
This theorem is referenced by: (None)
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