Theorem tskord 10606

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 5088	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝑇 ↔ 𝑦 ≺ 𝑇))
2	1	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇)))
3		eleq1 2825	. . . . 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 5088	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝑇 ↔ 𝐴 ≺ 𝑇))
6	5	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇)))
7		eleq1 2825	. . . . 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 774	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → 𝑇 ∈ Tarski)
10		onelss 6328	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		ssdomg 8836	. . . . . . . . . . . . 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 8952	. . . . . . . . . 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 3161	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) → (∀𝑦 ∈ 𝑥 ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇))
21		dfss3 3918	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑇 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇)
22		tskssel 10583	. . . . . . . . . . . 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 7747	. . 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 1540   ∈ wcel 2105  ∀wral 3062   ⊆ wss 3896   class class class wbr 5085  Oncon0 6286   ≼ cdom 8777   ≺ csdm 8778  Tarskictsk 10574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-ord 6289  df-on 6290  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-er 8544  df-en 8780  df-dom 8781  df-sdom 8782  df-tsk 10575

This theorem is referenced by:  tskcard  10607