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Theorem tskord 9804
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 4789 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑇𝑦𝑇))
21anbi2d 614 . . . . 5 (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦𝑇)))
3 eleq1 2838 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑇𝑦𝑇))
42, 3imbi12d 333 . . . 4 (𝑥 = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇)))
5 breq1 4789 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
65anbi2d 614 . . . . 5 (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴𝑇)))
7 eleq1 2838 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
86, 7imbi12d 333 . . . 4 (𝑥 = 𝐴 → (((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)))
9 simplrl 762 . . . . . . . . 9 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑇 ∈ Tarski)
10 onelss 5909 . . . . . . . . . . . . 13 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
11 ssdomg 8155 . . . . . . . . . . . . 13 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1210, 11syld 47 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1312imp 393 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
1413adantlr 694 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑦𝑥)
15 simplrr 763 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑥𝑇)
16 domsdomtr 8251 . . . . . . . . . 10 ((𝑦𝑥𝑥𝑇) → 𝑦𝑇)
1714, 15, 16syl2anc 573 . . . . . . . . 9 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑦𝑇)
18 pm2.27 42 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → (((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑦𝑇))
199, 17, 18syl2anc 573 . . . . . . . 8 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → (((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑦𝑇))
2019ralimdva 3111 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → ∀𝑦𝑥 𝑦𝑇))
21 dfss3 3741 . . . . . . . . . . 11 (𝑥𝑇 ↔ ∀𝑦𝑥 𝑦𝑇)
22 tskssel 9781 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑥𝑇) → 𝑥𝑇)
23223exp 1112 . . . . . . . . . . 11 (𝑇 ∈ Tarski → (𝑥𝑇 → (𝑥𝑇𝑥𝑇)))
2421, 23syl5bir 233 . . . . . . . . . 10 (𝑇 ∈ Tarski → (∀𝑦𝑥 𝑦𝑇 → (𝑥𝑇𝑥𝑇)))
2524com23 86 . . . . . . . . 9 (𝑇 ∈ Tarski → (𝑥𝑇 → (∀𝑦𝑥 𝑦𝑇𝑥𝑇)))
2625imp 393 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (∀𝑦𝑥 𝑦𝑇𝑥𝑇))
2726adantl 467 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 𝑦𝑇𝑥𝑇))
2820, 27syld 47 . . . . . 6 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑥𝑇))
2928ex 397 . . . . 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 7204 . . 3 (𝐴 ∈ On → ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇))
32313impib 1108 . 2 ((𝐴 ∈ On ∧ 𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
33323com12 1117 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wss 3723   class class class wbr 4786  Oncon0 5866  cdom 8107  csdm 8108  Tarskictsk 9772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-tsk 9773
This theorem is referenced by:  tskcard  9805
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