Theorem tskord 10202

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 5069	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝑇 ↔ 𝑦 ≺ 𝑇))
2	1	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇)))
3		eleq1 2900	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑇 ↔ 𝑦 ∈ 𝑇))
4	2, 3	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇) ↔ ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇)))
5		breq1 5069	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 𝑇 ↔ 𝐴 ≺ 𝑇))
6	5	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) ↔ (𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇)))
7		eleq1 2900	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑇 ↔ 𝐴 ∈ 𝑇))
8	6, 7	imbi12d 347	. . . 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 6233	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
11		ssdomg 8555	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → (𝑦 ⊆ 𝑥 → 𝑦 ≼ 𝑥))
12	10, 11	syld 47	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ≼ 𝑥))
13	12	imp 409	. . . . . . . . . . 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 8652	. . . . . . . . . 10 ⊢ ((𝑦 ≼ 𝑥 ∧ 𝑥 ≺ 𝑇) → 𝑦 ≺ 𝑇)
17	14, 15, 16	syl2anc 586	. . . . . . . . 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 586	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) ∧ 𝑦 ∈ 𝑥) → (((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑇))
20	19	ralimdva 3177	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇)) → (∀𝑦 ∈ 𝑥 ((𝑇 ∈ Tarski ∧ 𝑦 ≺ 𝑇) → 𝑦 ∈ 𝑇) → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇))
21		dfss3 3956	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑇 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇)
22		tskssel 10179	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ⊆ 𝑇 ∧ 𝑥 ≺ 𝑇) → 𝑥 ∈ 𝑇)
23	22	3exp 1115	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Tarski → (𝑥 ⊆ 𝑇 → (𝑥 ≺ 𝑇 → 𝑥 ∈ 𝑇)))
24	21, 23	syl5bir 245	. . . . . . . . . 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 409	. . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ≺ 𝑇) → (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑇 → 𝑥 ∈ 𝑇))
27	26	adantl 484	. . . . . . 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 415	. . . . 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 7572	. . 3 ⊢ (𝐴 ∈ On → ((𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇))
32	31	3impib 1112	. 2 ⊢ ((𝐴 ∈ On ∧ 𝑇 ∈ Tarski ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)
33	32	3com12 1119	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936   class class class wbr 5066  Oncon0 6191   ≼ cdom 8507   ≺ csdm 8508  Tarskictsk 10170

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-tsk 10171

This theorem is referenced by:  tskcard  10203