Theorem limensuci 3401

Description: A limit ordinal is equinumerous to its successor.

Hypothesis

Ref	Expression
limensuci.1	⊢ Lim A

Assertion

Ref	Expression
limensuci	⊢ (A ∈ B → A ≈ suc A)

Proof of Theorem limensuci

Step	Hyp	Ref	Expression
1		incom 1636	. . . . . 6 ⊢ ((A ∖ {∅}) ∩ {∅}) = ({∅} ∩ (A ∖ {∅}))
2		difdisj 1758	. . . . . 6 ⊢ ({∅} ∩ (A ∖ {∅})) = ∅
3	1, 2	eqtr 1119	. . . . 5 ⊢ ((A ∖ {∅}) ∩ {∅}) = ∅
4		limensuci.1	. . . . . . . 8 ⊢ Lim A
5		limord 2283	. . . . . . . 8 ⊢ (Lim A → Ord A)
6	4, 5	ax-mp 6	. . . . . . 7 ⊢ Ord A
7		ordeirr 2217	. . . . . . 7 ⊢ (Ord A → ¬ A ∈ A)
8	6, 7	ax-mp 6	. . . . . 6 ⊢ ¬ A ∈ A
9		disjsn 1836	. . . . . 6 ⊢ ((A ∩ {A}) = ∅ ↔ ¬ A ∈ A)
10	8, 9	mpbir 165	. . . . 5 ⊢ (A ∩ {A}) = ∅
11	3, 10	pm3.2i 234	. . . 4 ⊢ (((A ∖ {∅}) ∩ {∅}) = ∅ ∧ (A ∩ {A}) = ∅)
12		unen 3338	. . . 4 ⊢ ((((A ∖ {∅}) ≈ A ∧ {∅} ≈ {A}) ∧ (((A ∖ {∅}) ∩ {∅}) = ∅ ∧ (A ∩ {A}) = ∅)) → ((A ∖ {∅}) ∪ {∅}) ≈ (A ∪ {A}))
13	11, 12	mpan2 519	. . 3 ⊢ (((A ∖ {∅}) ≈ A ∧ {∅} ≈ {A}) → ((A ∖ {∅}) ∪ {∅}) ≈ (A ∪ {A}))
14		ensymg 3316	. . . 4 ⊢ ((A ∖ {∅}) ∈ V → (A ≈ (A ∖ {∅}) → (A ∖ {∅}) ≈ A))
15		difexg 1703	. . . 4 ⊢ (A ∈ B → (A ∖ {∅}) ∈ V)
16	4	limenpsi 3400	. . . 4 ⊢ (A ∈ B → A ≈ (A ∖ {∅}))
17	14, 15, 16	sylc 62	. . 3 ⊢ (A ∈ B → (A ∖ {∅}) ≈ A)
18		0ex 1745	. . . 4 ⊢ ∅ ∈ V
19		en2sn 3336	. . . 4 ⊢ ((∅ ∈ V ∧ A ∈ B) → {∅} ≈ {A})
20	18, 19	mpan 518	. . 3 ⊢ (A ∈ B → {∅} ≈ {A})
21	13, 17, 20	sylanc 361	. 2 ⊢ (A ∈ B → ((A ∖ {∅}) ∪ {∅}) ≈ (A ∪ {A}))
22		0ellim 2285	. . . . . 6 ⊢ (Lim A → ∅ ∈ A)
23	4, 22	ax-mp 6	. . . . 5 ⊢ ∅ ∈ A
24	18	snss 1849	. . . . 5 ⊢ (∅ ∈ A ↔ {∅} ⊆ A)
25	23, 24	mpbi 164	. . . 4 ⊢ {∅} ⊆ A
26		ssundif 1764	. . . 4 ⊢ ({∅} ⊆ A ↔ ({∅} ∪ (A ∖ {∅})) = A)
27	25, 26	mpbi 164	. . 3 ⊢ ({∅} ∪ (A ∖ {∅})) = A
28		uncom 1604	. . 3 ⊢ ({∅} ∪ (A ∖ {∅})) = ((A ∖ {∅}) ∪ {∅})
29	27, 28	eqtr3 1121	. 2 ⊢ A = ((A ∖ {∅}) ∪ {∅})
30		df-suc 2205	. 2 ⊢ suc A = (A ∪ {A})
31	21, 29, 30	3brtr4g 2088	1 ⊢ (A ∈ B → A ≈ suc A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  {csn 1808   class class class wbr 2054  Ord word 2198  Lim wlim 2200  suc csuc 2201   ≈ cen 3271

This theorem is referenced by:  limensuc 3402  omensuc 3483

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-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  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-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-1o 3104  df-er 3200  df-en 3274  df-dom 3275

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