Theorem tskord 10821

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 ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)

Proof of Theorem tskord

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5145	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝑇 ↔ 𝑦 ≺ 𝑇))
2	1	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇)))
3		eleq1 2828	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑇 ↔ 𝑦 ∈ 𝑇))
4	2, 3	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇)))
5		breq1 5145	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝑇 ↔ 𝐴 ≺ 𝑇))
6	5	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇)))
7		eleq1 2828	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑇 ↔ 𝐴 ∈ 𝑇))
8	6, 7	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)))
9		simplrl 776	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑇 ∈ Tarski)
10		onelss 6425	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		ssdomg 9041	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (𝑦 ⊆ 𝑥 → 𝑦 ≼ 𝑥))
12	10, 11	syld 47	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ≼ 𝑥))
13	12	imp 406	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ≼ 𝑥)
14	13	adantlr 715	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ≼ 𝑥)
15		simplrr 777	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑥 ≺ 𝑇)
16		domsdomtr 9153	. . . . . . . . . 10 ⊢ ((𝑦 ≼ 𝑥 ∧ 𝑥 ≺ 𝑇) → 𝑦 ≺ 𝑇)
17	14, 15, 16	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ≺ 𝑇)
18		pm2.27 42	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → (((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑇))
19	9, 17, 18	syl2anc 584	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → (((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑇))
20	19	ralimdva 3166	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) → (∀𝑦 ∈ 𝑥 ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇))
21		dfss3 3971	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑇 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇)
22		tskssel 10798	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ⊆ 𝑇 ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇)
23	22	3exp 1119	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Tarski → (𝑥 ⊆ 𝑇 → (𝑥 ≺ 𝑇 → 𝑥 ∈ 𝑇)))
24	21, 23	biimtrrid 243	. . . . . . . . . 10 ⊢ (𝑇 ∈ Tarski → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇 → (𝑥 ≺ 𝑇 → 𝑥 ∈ 𝑇)))
25	24	com23 86	. . . . . . . . 9 ⊢ (𝑇 ∈ Tarski → (𝑥 ≺ 𝑇 → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇 → 𝑥 ∈ 𝑇)))
26	25	imp 406	. . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇 → 𝑥 ∈ 𝑇))
27	26	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇 → 𝑥 ∈ 𝑇))
28	20, 27	syld 47	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) → (∀𝑦 ∈ 𝑥 ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇))
29	28	ex 412	. . . . 5 ⊢ (𝑥 ∈ On → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → (∀𝑦 ∈ 𝑥 ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)))
30	29	com23 86	. . . 4 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇)))
31	4, 8, 30	tfis3 7880	. . 3 ⊢ (𝐴 ∈ On → ((𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇))
32	31	3impib 1116	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)
33	32	3com12 1123	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060   ⊆ wss 3950   class class class wbr 5142  Oncon0 6383   ≼ cdom 8984   ≺ csdm 8985  Tarskictsk 10789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-tsk 10790

This theorem is referenced by:  tskcard  10822