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Theorem tskord 9587
Description: A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)
Assertion
Ref Expression
tskord ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)

Proof of Theorem tskord
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4647 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑇𝑦𝑇))
21anbi2d 739 . . . . 5 (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦𝑇)))
3 eleq1 2687 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑇𝑦𝑇))
42, 3imbi12d 334 . . . 4 (𝑥 = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇)))
5 breq1 4647 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
65anbi2d 739 . . . . 5 (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴𝑇)))
7 eleq1 2687 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
86, 7imbi12d 334 . . . 4 (𝑥 = 𝐴 → (((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)))
9 simplrl 799 . . . . . . . . 9 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑇 ∈ Tarski)
10 onelss 5754 . . . . . . . . . . . . 13 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
11 ssdomg 7986 . . . . . . . . . . . . 13 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1210, 11syld 47 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1312imp 445 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
1413adantlr 750 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑦𝑥)
15 simplrr 800 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑥𝑇)
16 domsdomtr 8080 . . . . . . . . . 10 ((𝑦𝑥𝑥𝑇) → 𝑦𝑇)
1714, 15, 16syl2anc 692 . . . . . . . . 9 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑦𝑇)
18 pm2.27 42 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → (((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑦𝑇))
199, 17, 18syl2anc 692 . . . . . . . 8 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → (((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑦𝑇))
2019ralimdva 2959 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → ∀𝑦𝑥 𝑦𝑇))
21 dfss3 3585 . . . . . . . . . . 11 (𝑥𝑇 ↔ ∀𝑦𝑥 𝑦𝑇)
22 tskssel 9564 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑥𝑇) → 𝑥𝑇)
23223exp 1262 . . . . . . . . . . 11 (𝑇 ∈ Tarski → (𝑥𝑇 → (𝑥𝑇𝑥𝑇)))
2421, 23syl5bir 233 . . . . . . . . . 10 (𝑇 ∈ Tarski → (∀𝑦𝑥 𝑦𝑇 → (𝑥𝑇𝑥𝑇)))
2524com23 86 . . . . . . . . 9 (𝑇 ∈ Tarski → (𝑥𝑇 → (∀𝑦𝑥 𝑦𝑇𝑥𝑇)))
2625imp 445 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (∀𝑦𝑥 𝑦𝑇𝑥𝑇))
2726adantl 482 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 𝑦𝑇𝑥𝑇))
2820, 27syld 47 . . . . . 6 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑥𝑇))
2928ex 450 . . . . 5 (𝑥 ∈ On → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑥𝑇)))
3029com23 86 . . . 4 (𝑥 ∈ On → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇)))
314, 8, 30tfis3 7042 . . 3 (𝐴 ∈ On → ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇))
32313impib 1260 . 2 ((𝐴 ∈ On ∧ 𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
33323com12 1267 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wss 3567   class class class wbr 4644  Oncon0 5711  cdom 7938  csdm 7939  Tarskictsk 9555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-tsk 9556
This theorem is referenced by:  tskcard  9588
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