Theorem tskord 10536

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 5077	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝑇 ↔ 𝑦 ≺ 𝑇))
2	1	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇)))
3		eleq1 2826	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑇 ↔ 𝑦 ∈ 𝑇))
4	2, 3	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇)))
5		breq1 5077	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝑇 ↔ 𝐴 ≺ 𝑇))
6	5	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇)))
7		eleq1 2826	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑇 ↔ 𝐴 ∈ 𝑇))
8	6, 7	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)))
9		simplrl 774	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑇 ∈ Tarski)
10		onelss 6308	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		ssdomg 8786	. . . . . . . . . . . . 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 712	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ≼ 𝑥)
15		simplrr 775	. . . . . . . . . 10 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑥 ≺ 𝑇)
16		domsdomtr 8899	. . . . . . . . . 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 3108	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) → (∀𝑦 ∈ 𝑥 ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇))
21		dfss3 3909	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑇 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇)
22		tskssel 10513	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ⊆ 𝑇 ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇)
23	22	3exp 1118	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Tarski → (𝑥 ⊆ 𝑇 → (𝑥 ≺ 𝑇 → 𝑥 ∈ 𝑇)))
24	21, 23	syl5bir 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 7704	. . 3 ⊢ (𝐴 ∈ On → ((𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇))
32	31	3impib 1115	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)
33	32	3com12 1122	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887   class class class wbr 5074  Oncon0 6266   ≼ cdom 8731   ≺ csdm 8732  Tarskictsk 10504

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-tsk 10505

This theorem is referenced by:  tskcard  10537