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Theorem ttac 43025
Description: Tarski's theorem about choice: infxpidm 10600 is equivalent to ax-ac 10497. (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 10176 . 2 (CHOICE ↔ dom card = V)
2 vex 3482 . . . . . 6 𝑐 ∈ V
3 eleq2 2828 . . . . . 6 (dom card = V → (𝑐 ∈ dom card ↔ 𝑐 ∈ V))
42, 3mpbiri 258 . . . . 5 (dom card = V → 𝑐 ∈ dom card)
5 infxpidm2 10055 . . . . . 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 1925 . . 3 (dom card = V → ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
9 finnum 9986 . . . . . . 7 (𝑎 ∈ Fin → 𝑎 ∈ dom card)
109adantl 481 . . . . . 6 ((∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) ∧ 𝑎 ∈ Fin) → 𝑎 ∈ dom card)
11 harcl 9597 . . . . . . . . 9 (har‘𝑎) ∈ On
12 onenon 9987 . . . . . . . . 9 ((har‘𝑎) ∈ On → (har‘𝑎) ∈ dom card)
1311, 12ax-mp 5 . . . . . . . 8 (har‘𝑎) ∈ dom card
14 fvex 6920 . . . . . . . . . . . . . 14 (har‘𝑎) ∈ V
15 vex 3482 . . . . . . . . . . . . . 14 𝑎 ∈ V
1614, 15unex 7763 . . . . . . . . . . . . 13 ((har‘𝑎) ∪ 𝑎) ∈ V
17 harinf 43023 . . . . . . . . . . . . . . 15 ((𝑎 ∈ V ∧ ¬ 𝑎 ∈ Fin) → ω ⊆ (har‘𝑎))
1815, 17mpan 690 . . . . . . . . . . . . . 14 𝑎 ∈ Fin → ω ⊆ (har‘𝑎))
19 ssun1 4188 . . . . . . . . . . . . . 14 (har‘𝑎) ⊆ ((har‘𝑎) ∪ 𝑎)
2018, 19sstrdi 4008 . . . . . . . . . . . . 13 𝑎 ∈ Fin → ω ⊆ ((har‘𝑎) ∪ 𝑎))
21 ssdomg 9039 . . . . . . . . . . . . 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 5152 . . . . . . . . . . . . . 14 (𝑐 = ((har‘𝑎) ∪ 𝑎) → (ω ≼ 𝑐 ↔ ω ≼ ((har‘𝑎) ∪ 𝑎)))
24 xpeq12 5714 . . . . . . . . . . . . . . . 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 5161 . . . . . . . . . . . . . 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 3605 . . . . . . . . . . . 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 9600 . . . . . . . . . . . 12 ¬ (har‘𝑎) ≼ 𝑎
33 ssdomg 9039 . . . . . . . . . . . . . 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 9046 . . . . . . . . . . . . 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 9626 . . . . . . . . . . 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 10102 . . . . . . . . . 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 10076 . . . . . . . 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 4189 . . . . . . 7 𝑎 ⊆ ((har‘𝑎) ∪ 𝑎)
48 ssnum 10077 . . . . . . 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 813 . . . . 5 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → 𝑎 ∈ dom card)
5150alrimiv 1925 . . . 4 (∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐) → ∀𝑎 𝑎 ∈ dom card)
52 eqv 3488 . . . 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 1535   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  wss 3963   class class class wbr 5148   × cxp 5687  dom cdm 5689  Oncon0 6386  cfv 6563  ωcom 7887  cen 8981  cdom 8982  Fincfn 8984  harchar 9594  * cwdom 9602  cardccrd 9973  CHOICEwac 10153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-har 9595  df-wdom 9603  df-card 9977  df-acn 9980  df-ac 10154
This theorem is referenced by: (None)
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