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Theorem tskord 10724
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 5112 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑇𝑦𝑇))
21anbi2d 630 . . . . 5 (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦𝑇)))
3 eleq1 2822 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑇𝑦𝑇))
42, 3imbi12d 345 . . . 4 (𝑥 = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇)))
5 breq1 5112 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
65anbi2d 630 . . . . 5 (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴𝑇)))
7 eleq1 2822 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
86, 7imbi12d 345 . . . 4 (𝑥 = 𝐴 → (((𝑇 ∈ Tarski ∧ 𝑥𝑇) → 𝑥𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)))
9 simplrl 776 . . . . . . . . 9 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑇 ∈ Tarski)
10 onelss 6363 . . . . . . . . . . . . 13 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
11 ssdomg 8946 . . . . . . . . . . . . 13 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1210, 11syld 47 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
1312imp 408 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
1413adantlr 714 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑦𝑥)
15 simplrr 777 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑥𝑇)
16 domsdomtr 9062 . . . . . . . . . 10 ((𝑦𝑥𝑥𝑇) → 𝑦𝑇)
1714, 15, 16syl2anc 585 . . . . . . . . 9 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → 𝑦𝑇)
18 pm2.27 42 . . . . . . . . 9 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → (((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑦𝑇))
199, 17, 18syl2anc 585 . . . . . . . 8 (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) ∧ 𝑦𝑥) → (((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑦𝑇))
2019ralimdva 3161 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → ∀𝑦𝑥 𝑦𝑇))
21 dfss3 3936 . . . . . . . . . . 11 (𝑥𝑇 ↔ ∀𝑦𝑥 𝑦𝑇)
22 tskssel 10701 . . . . . . . . . . . 12 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑥𝑇) → 𝑥𝑇)
23223exp 1120 . . . . . . . . . . 11 (𝑇 ∈ Tarski → (𝑥𝑇 → (𝑥𝑇𝑥𝑇)))
2421, 23biimtrrid 242 . . . . . . . . . 10 (𝑇 ∈ Tarski → (∀𝑦𝑥 𝑦𝑇 → (𝑥𝑇𝑥𝑇)))
2524com23 86 . . . . . . . . 9 (𝑇 ∈ Tarski → (𝑥𝑇 → (∀𝑦𝑥 𝑦𝑇𝑥𝑇)))
2625imp 408 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → (∀𝑦𝑥 𝑦𝑇𝑥𝑇))
2726adantl 483 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 𝑦𝑇𝑥𝑇))
2820, 27syld 47 . . . . . 6 ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥𝑇)) → (∀𝑦𝑥 ((𝑇 ∈ Tarski ∧ 𝑦𝑇) → 𝑦𝑇) → 𝑥𝑇))
2928ex 414 . . . . 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 7798 . . 3 (𝐴 ∈ On → ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇))
32313impib 1117 . 2 ((𝐴 ∈ On ∧ 𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
33323com12 1124 1 ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wss 3914   class class class wbr 5109  Oncon0 6321  cdom 8887  csdm 8888  Tarskictsk 10692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-tsk 10693
This theorem is referenced by:  tskcard  10725
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