Theorem tskord 10774

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 5151	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝑇 ↔ 𝑦 ≺ 𝑇))
2	1	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇)))
3		eleq1 2821	. . . . 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 5151	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝑇 ↔ 𝐴 ≺ 𝑇))
6	5	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇)))
7		eleq1 2821	. . . . 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 775	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑇 ∈ Tarski)
10		onelss 6406	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		ssdomg 8995	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (𝑦 ⊆ 𝑥 → 𝑦 ≼ 𝑥))
12	10, 11	syld 47	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ≼ 𝑥))
13	12	imp 407	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ≼ 𝑥)
14	13	adantlr 713	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ≼ 𝑥)
15		simplrr 776	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑥 ≺ 𝑇)
16		domsdomtr 9111	. . . . . . . . . 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 3167	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) → (∀𝑦 ∈ 𝑥 ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇))
21		dfss3 3970	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑇 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇)
22		tskssel 10751	. . . . . . . . . . . 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 242	. . . . . . . . . 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 407	. . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇 → 𝑥 ∈ 𝑇))
27	26	adantl 482	. . . . . . 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 413	. . . . 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 7846	. . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3948   class class class wbr 5148  Oncon0 6364   ≼ cdom 8936   ≺ csdm 8937  Tarskictsk 10742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-tsk 10743

This theorem is referenced by:  tskcard  10775