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Theorem tskord 9481
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 4586 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑇𝑦𝑇))
21anbi2d 736 . . . . 5 (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦𝑇)))
3 eleq1 2676 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑇𝑦𝑇))
42, 3imbi12d 333 . . . 4 (𝑥 = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇)))
5 breq1 4586 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
65anbi2d 736 . . . . 5 (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴𝑇)))
7 eleq1 2676 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
86, 7imbi12d 333 . . . 4 (𝑥 = 𝐴 → (((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)))
9 simplrl 796 . . . . . . . . 9 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑇 ∈ Tarski)
10 onelss 5683 . . . . . . . . . . . . 13 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
11 ssdomg 7887 . . . . . . . . . . . . 13 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1210, 11syld 46 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1312imp 444 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
1413adantlr 747 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑦𝑥)
15 simplrr 797 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑥𝑇)
16 domsdomtr 7980 . . . . . . . . . 10 ((𝑦𝑥𝑥𝑇) → 𝑦𝑇)
1714, 15, 16syl2anc 691 . . . . . . . . 9 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑦𝑇)
18 pm2.27 41 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → (((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑦𝑇))
199, 17, 18syl2anc 691 . . . . . . . 8 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → (((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑦𝑇))
2019ralimdva 2945 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → ∀𝑦𝑥 𝑦𝑇))
21 dfss3 3558 . . . . . . . . . . 11 (𝑥𝑇 ↔ ∀𝑦𝑥 𝑦𝑇)
22 tskssel 9458 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑥𝑇) → 𝑥𝑇)
23223exp 1256 . . . . . . . . . . 11 (𝑇 ∈ Tarski → (𝑥𝑇 → (𝑥𝑇𝑥𝑇)))
2421, 23syl5bir 232 . . . . . . . . . 10 (𝑇 ∈ Tarski → (∀𝑦𝑥 𝑦𝑇 → (𝑥𝑇𝑥𝑇)))
2524com23 84 . . . . . . . . 9 (𝑇 ∈ Tarski → (𝑥𝑇 → (∀𝑦𝑥 𝑦𝑇𝑥𝑇)))
2625imp 444 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (∀𝑦𝑥 𝑦𝑇𝑥𝑇))
2726adantl 481 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 𝑦𝑇𝑥𝑇))
2820, 27syld 46 . . . . . 6 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑥𝑇))
2928ex 449 . . . . 5 (𝑥 ∈ On → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑥𝑇)))
3029com23 84 . . . 4 (𝑥 ∈ On → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇)))
314, 8, 30tfis3 6949 . . 3 (𝐴 ∈ On → ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇))
32313impib 1254 . 2 ((𝐴 ∈ On ∧ 𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
33323com12 1261 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wss 3540   class class class wbr 4583  Oncon0 5640  cdom 7839  csdm 7840  Tarskictsk 9449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-tsk 9450
This theorem is referenced by:  tskcard  9482
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