Theorem onomeneq 3414

Description: An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse.

Assertion

Ref	Expression
onomeneq	⊢ ((A ∈ On ∧ B ∈ ω) → (A ≈ B ↔ A = B))

Proof of Theorem onomeneq

Step	Hyp	Ref	Expression
1		nneneq 3408	. . . . 5 ⊢ ((A ∈ ω ∧ B ∈ ω) → (A ≈ B ↔ A = B))
2	1	biimpa 324	. . . 4 ⊢ (((A ∈ ω ∧ B ∈ ω) ∧ A ≈ B) → A = B)
3		php5 3413	. . . . . . . . . 10 ⊢ (B ∈ ω → ¬ B ≈ suc B)
4	3	adantr 306	. . . . . . . . 9 ⊢ ((B ∈ ω ∧ A ≈ B) → ¬ B ≈ suc B)
5		enen1 3375	. . . . . . . . 9 ⊢ ((B ∈ ω ∧ A ≈ B) → (A ≈ suc B ↔ B ≈ suc B))
6	4, 5	mtbird 537	. . . . . . . 8 ⊢ ((B ∈ ω ∧ A ≈ B) → ¬ A ≈ suc B)
7	6	adantll 309	. . . . . . 7 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → ¬ A ≈ suc B)
8		endomtr 3325	. . . . . . . . . . . . 13 ⊢ ((A ≈ B ∧ B ≼ suc B) → A ≼ suc B)
9		sssucid 2300	. . . . . . . . . . . . . 14 ⊢ B ⊆ suc B
10		ssdomg 3311	. . . . . . . . . . . . . 14 ⊢ (B ∈ ω → (B ⊆ suc B → B ≼ suc B))
11	9, 10	mpi 44	. . . . . . . . . . . . 13 ⊢ (B ∈ ω → B ≼ suc B)
12	8, 11	sylan2 346	. . . . . . . . . . . 12 ⊢ ((A ≈ B ∧ B ∈ ω) → A ≼ suc B)
13	12	ancoms 334	. . . . . . . . . . 11 ⊢ ((B ∈ ω ∧ A ≈ B) → A ≼ suc B)
14	13	a1d 14	. . . . . . . . . 10 ⊢ ((B ∈ ω ∧ A ≈ B) → (ω ⊆ A → A ≼ suc B))
15	14	adantll 309	. . . . . . . . 9 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → (ω ⊆ A → A ≼ suc B))
16		ssel 1502	. . . . . . . . . . . . . . 15 ⊢ (ω ⊆ A → (B ∈ ω → B ∈ A))
17	16	com12 13	. . . . . . . . . . . . . 14 ⊢ (B ∈ ω → (ω ⊆ A → B ∈ A))
18	17	adantr 306	. . . . . . . . . . . . 13 ⊢ ((B ∈ ω ∧ A ∈ On) → (ω ⊆ A → B ∈ A))
19		ordelsuc 2322	. . . . . . . . . . . . . 14 ⊢ ((B ∈ ω ∧ Ord A) → (B ∈ A ↔ suc B ⊆ A))
20		eloni 2209	. . . . . . . . . . . . . 14 ⊢ (A ∈ On → Ord A)
21	19, 20	sylan2 346	. . . . . . . . . . . . 13 ⊢ ((B ∈ ω ∧ A ∈ On) → (B ∈ A ↔ suc B ⊆ A))
22	18, 21	sylibd 177	. . . . . . . . . . . 12 ⊢ ((B ∈ ω ∧ A ∈ On) → (ω ⊆ A → suc B ⊆ A))
23		ssdom2g 3312	. . . . . . . . . . . . 13 ⊢ (A ∈ On → (suc B ⊆ A → suc B ≼ A))
24	23	adantl 305	. . . . . . . . . . . 12 ⊢ ((B ∈ ω ∧ A ∈ On) → (suc B ⊆ A → suc B ≼ A))
25	22, 24	syld 27	. . . . . . . . . . 11 ⊢ ((B ∈ ω ∧ A ∈ On) → (ω ⊆ A → suc B ≼ A))
26	25	ancoms 334	. . . . . . . . . 10 ⊢ ((A ∈ On ∧ B ∈ ω) → (ω ⊆ A → suc B ≼ A))
27	26	adantr 306	. . . . . . . . 9 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → (ω ⊆ A → suc B ≼ A))
28	15, 27	jcad 455	. . . . . . . 8 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → (ω ⊆ A → (A ≼ suc B ∧ suc B ≼ A)))
29		sbth 3359	. . . . . . . 8 ⊢ ((A ≼ suc B ∧ suc B ≼ A) → A ≈ suc B)
30	28, 29	syl6 23	. . . . . . 7 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → (ω ⊆ A → A ≈ suc B))
31	7, 30	mtod 95	. . . . . 6 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → ¬ ω ⊆ A)
32		ordom 2382	. . . . . . . . . 10 ⊢ Ord ω
33		ordtri1 2231	. . . . . . . . . 10 ⊢ ((Ord ω ∧ Ord A) → (ω ⊆ A ↔ ¬ A ∈ ω))
34	32, 33	mpan 518	. . . . . . . . 9 ⊢ (Ord A → (ω ⊆ A ↔ ¬ A ∈ ω))
35	20, 34	syl 12	. . . . . . . 8 ⊢ (A ∈ On → (ω ⊆ A ↔ ¬ A ∈ ω))
36	35	bicon2d 404	. . . . . . 7 ⊢ (A ∈ On → (A ∈ ω ↔ ¬ ω ⊆ A))
37	36	ad2antll 320	. . . . . 6 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → (A ∈ ω ↔ ¬ ω ⊆ A))
38	31, 37	mpbird 171	. . . . 5 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → A ∈ ω)
39		pm3.27 260	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ ω) → B ∈ ω)
40	39	adantr 306	. . . . 5 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → B ∈ ω)
41	38, 40	jca 236	. . . 4 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → (A ∈ ω ∧ B ∈ ω))
42		pm3.27 260	. . . 4 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → A ≈ B)
43	2, 41, 42	sylanc 361	. . 3 ⊢ (((A ∈ On ∧ B ∈ ω) ∧ A ≈ B) → A = B)
44	43	exp 291	. 2 ⊢ ((A ∈ On ∧ B ∈ ω) → (A ≈ B → A = B))
45		eqeng 3296	. . 3 ⊢ (A ∈ On → (A = B → A ≈ B))
46	45	adantr 306	. 2 ⊢ ((A ∈ On ∧ B ∈ ω) → (A = B → A ≈ B))
47	44, 46	impbid 397	1 ⊢ ((A ∈ On ∧ B ∈ ω) → (A ≈ B ↔ A = B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487   class class class wbr 2054  Ord word 2198  Oncon0 2199  suc csuc 2201  ωcom 2372   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  onfin 3415

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

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-v 1349  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-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-er 3200  df-en 3274  df-dom 3275  df-sdom 3276

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