Theorem onomeneq 4525

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		php5 4524	. . . . . . . . . 10 ⊢ (B ∈ ω → ¬ B ≈ suc B)
2	1	adantr 391	. . . . . . . . 9 ⊢ ((B ∈ ω ⋀ A ≈ B) → ¬ B ≈ suc B)
3		enen1 4483	. . . . . . . . 9 ⊢ ((B ∈ ω ⋀ A ≈ B) → (A ≈ suc B ↔ B ≈ suc B))
4	2, 3	mtbird 717	. . . . . . . 8 ⊢ ((B ∈ ω ⋀ A ≈ B) → ¬ A ≈ suc B)
5	4	adantll 394	. . . . . . 7 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → ¬ A ≈ suc B)
6		endomtr 4426	. . . . . . . . . . . . 13 ⊢ ((A ≈ B ⋀ B ≼ suc B) → A ≼ suc B)
7		sssucid 3053	. . . . . . . . . . . . . 14 ⊢ B ⊆ suc B
8		ssdomg 4414	. . . . . . . . . . . . . 14 ⊢ (B ∈ ω → (B ⊆ suc B → B ≼ suc B))
9	7, 8	mpi 44	. . . . . . . . . . . . 13 ⊢ (B ∈ ω → B ≼ suc B)
10	6, 9	sylan2 453	. . . . . . . . . . . 12 ⊢ ((A ≈ B ⋀ B ∈ ω) → A ≼ suc B)
11	10	ancoms 438	. . . . . . . . . . 11 ⊢ ((B ∈ ω ⋀ A ≈ B) → A ≼ suc B)
12	11	a1d 12	. . . . . . . . . 10 ⊢ ((B ∈ ω ⋀ A ≈ B) → (ω ⊆ A → A ≼ suc B))
13	12	adantll 394	. . . . . . . . 9 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → (ω ⊆ A → A ≼ suc B))
14		ssel 2066	. . . . . . . . . . . . . . 15 ⊢ (ω ⊆ A → (B ∈ ω → B ∈ A))
15	14	com12 11	. . . . . . . . . . . . . 14 ⊢ (B ∈ ω → (ω ⊆ A → B ∈ A))
16	15	adantr 391	. . . . . . . . . . . . 13 ⊢ ((B ∈ ω ⋀ A ∈ On) → (ω ⊆ A → B ∈ A))
17		ordelsuc 3077	. . . . . . . . . . . . . 14 ⊢ ((B ∈ ω ⋀ Ord A) → (B ∈ A ↔ suc B ⊆ A))
18		eloni 2964	. . . . . . . . . . . . . 14 ⊢ (A ∈ On → Ord A)
19	17, 18	sylan2 453	. . . . . . . . . . . . 13 ⊢ ((B ∈ ω ⋀ A ∈ On) → (B ∈ A ↔ suc B ⊆ A))
20	16, 19	sylibd 202	. . . . . . . . . . . 12 ⊢ ((B ∈ ω ⋀ A ∈ On) → (ω ⊆ A → suc B ⊆ A))
21		ssdom2g 4415	. . . . . . . . . . . . 13 ⊢ (A ∈ On → (suc B ⊆ A → suc B ≼ A))
22	21	adantl 390	. . . . . . . . . . . 12 ⊢ ((B ∈ ω ⋀ A ∈ On) → (suc B ⊆ A → suc B ≼ A))
23	20, 22	syld 27	. . . . . . . . . . 11 ⊢ ((B ∈ ω ⋀ A ∈ On) → (ω ⊆ A → suc B ≼ A))
24	23	ancoms 438	. . . . . . . . . 10 ⊢ ((A ∈ On ⋀ B ∈ ω) → (ω ⊆ A → suc B ≼ A))
25	24	adantr 391	. . . . . . . . 9 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → (ω ⊆ A → suc B ≼ A))
26	13, 25	jcad 602	. . . . . . . 8 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → (ω ⊆ A → (A ≼ suc B ⋀ suc B ≼ A)))
27		sbth 4463	. . . . . . . 8 ⊢ ((A ≼ suc B ⋀ suc B ≼ A) → A ≈ suc B)
28	26, 27	syl6 22	. . . . . . 7 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → (ω ⊆ A → A ≈ suc B))
29	5, 28	mtod 108	. . . . . 6 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → ¬ ω ⊆ A)
30		ordom 3147	. . . . . . . . . 10 ⊢ Ord ω
31		ordtri1 2986	. . . . . . . . . 10 ⊢ ((Ord ω ⋀ Ord A) → (ω ⊆ A ↔ ¬ A ∈ ω))
32	30, 31	mpan 697	. . . . . . . . 9 ⊢ (Ord A → (ω ⊆ A ↔ ¬ A ∈ ω))
33	18, 32	syl 10	. . . . . . . 8 ⊢ (A ∈ On → (ω ⊆ A ↔ ¬ A ∈ ω))
34	33	con2bid 528	. . . . . . 7 ⊢ (A ∈ On → (A ∈ ω ↔ ¬ ω ⊆ A))
35	34	ad2antrr 406	. . . . . 6 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → (A ∈ ω ↔ ¬ ω ⊆ A))
36	29, 35	mpbird 196	. . . . 5 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → A ∈ ω)
37		simplr 415	. . . . 5 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → B ∈ ω)
38	36, 37	jca 288	. . . 4 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → (A ∈ ω ⋀ B ∈ ω))
39		nneneq 4518	. . . . 5 ⊢ ((A ∈ ω ⋀ B ∈ ω) → (A ≈ B ↔ A = B))
40	39	biimpa 418	. . . 4 ⊢ (((A ∈ ω ⋀ B ∈ ω) ⋀ A ≈ B) → A = B)
41	38, 40	sylancom 477	. . 3 ⊢ (((A ∈ On ⋀ B ∈ ω) ⋀ A ≈ B) → A = B)
42	41	ex 373	. 2 ⊢ ((A ∈ On ⋀ B ∈ ω) → (A ≈ B → A = B))
43		eqeng 4398	. . 3 ⊢ (A ∈ On → (A = B → A ≈ B))
44	43	adantr 391	. 2 ⊢ ((A ∈ On ⋀ B ∈ ω) → (A = B → A ≈ B))
45	42, 44	impbid 518	1 ⊢ ((A ∈ On ⋀ B ∈ ω) → (A ≈ B ↔ A = B))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960   ⊆ wss 2050   class class class wbr 2624  Ord word 2953  Oncon0 2954  suc csuc 2956  ωcom 3137   ≈ cen 4370   ≼ cdom 4371

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376

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