Theorem alephord 3680

Description: Ordering property of the aleph function.

Assertion

Ref	Expression
alephord	⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B ↔ (ℵ ‘A) ≺ (ℵ ‘B)))

Proof of Theorem alephord

Step	Hyp	Ref	Expression
1		alephordi 3679	. . 3 ⊢ (B ∈ On → (A ∈ B → (ℵ ‘A) ≺ (ℵ ‘B)))
2	1	adantl 305	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B → (ℵ ‘A) ≺ (ℵ ‘B)))
3		alephordi 3679	. . . . . . . . 9 ⊢ (A ∈ On → (B ∈ A → (ℵ ‘B) ≺ (ℵ ‘A)))
4	3	con3d 87	. . . . . . . 8 ⊢ (A ∈ On → (¬ (ℵ ‘B) ≺ (ℵ ‘A) → ¬ B ∈ A))
5		alephon 3671	. . . . . . . . 9 ⊢ (ℵ ‘A) ∈ On
6		alephon 3671	. . . . . . . . 9 ⊢ (ℵ ‘B) ∈ On
7		domtri 3644	. . . . . . . . 9 ⊢ (((ℵ ‘A) ∈ On ∧ (ℵ ‘B) ∈ On) → ((ℵ ‘A) ≼ (ℵ ‘B) ↔ ¬ (ℵ ‘B) ≺ (ℵ ‘A)))
8	5, 6, 7	mp2an 520	. . . . . . . 8 ⊢ ((ℵ ‘A) ≼ (ℵ ‘B) ↔ ¬ (ℵ ‘B) ≺ (ℵ ‘A))
9	4, 8	syl5ib 181	. . . . . . 7 ⊢ (A ∈ On → ((ℵ ‘A) ≼ (ℵ ‘B) → ¬ B ∈ A))
10	9	adantr 306	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ On) → ((ℵ ‘A) ≼ (ℵ ‘B) → ¬ B ∈ A))
11		ontri1 2232	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ On) → (A ⊆ B ↔ ¬ B ∈ A))
12	10, 11	sylibrd 179	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On) → ((ℵ ‘A) ≼ (ℵ ‘B) → A ⊆ B))
13		fveq2 2832	. . . . . . . 8 ⊢ (A = B → (ℵ ‘A) = (ℵ ‘B))
14		eqeng 3296	. . . . . . . . 9 ⊢ ((ℵ ‘A) ∈ On → ((ℵ ‘A) = (ℵ ‘B) → (ℵ ‘A) ≈ (ℵ ‘B)))
15	5, 14	ax-mp 6	. . . . . . . 8 ⊢ ((ℵ ‘A) = (ℵ ‘B) → (ℵ ‘A) ≈ (ℵ ‘B))
16	13, 15	syl 12	. . . . . . 7 ⊢ (A = B → (ℵ ‘A) ≈ (ℵ ‘B))
17	16	con3i 90	. . . . . 6 ⊢ (¬ (ℵ ‘A) ≈ (ℵ ‘B) → ¬ A = B)
18	17	a1i 7	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On) → (¬ (ℵ ‘A) ≈ (ℵ ‘B) → ¬ A = B))
19	12, 18	anim12d 431	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (((ℵ ‘A) ≼ (ℵ ‘B) ∧ ¬ (ℵ ‘A) ≈ (ℵ ‘B)) → (A ⊆ B ∧ ¬ A = B)))
20		onelpsst 2253	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B ↔ (A ⊆ B ∧ ¬ A = B)))
21	19, 20	sylibrd 179	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → (((ℵ ‘A) ≼ (ℵ ‘B) ∧ ¬ (ℵ ‘A) ≈ (ℵ ‘B)) → A ∈ B))
22		brsdom 3286	. . 3 ⊢ ((ℵ ‘A) ≺ (ℵ ‘B) ↔ ((ℵ ‘A) ≼ (ℵ ‘B) ∧ ¬ (ℵ ‘A) ≈ (ℵ ‘B)))
23	21, 22	syl5ib 181	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → ((ℵ ‘A) ≺ (ℵ ‘B) → A ∈ B))
24	2, 23	impbid 397	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B ↔ (ℵ ‘A) ≺ (ℵ ‘B)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487   class class class wbr 2054  Oncon0 2199   ‘cfv 2422   ≈ cen 3271   ≼ cdom 3272   ≺ csdm 3273  ℵcale 3621

This theorem is referenced by:  alephord2 3681

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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