Theorem onmindif2 3118

Description: The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed.

Assertion

Ref	Expression
onmindif2	|- ((A (_ On /\ A =/= (/)) -> |^|A e. |^|(A \ {|^|A}))

Proof of Theorem onmindif2

Step	Hyp	Ref	Expression
1		onnmin 3072	. . . . . . . . . 10 |- ((A (_ On /\ x e. A) -> -. x e. |^|A)
2	1	adantlr 402	. . . . . . . . 9 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> -. x e. |^|A)
3		ontri1 3038	. . . . . . . . . . 11 |- ((|^|A e. On /\ x e. On) -> (|^|A (_ x <-> -. x e. |^|A))
4		onsseleq 3056	. . . . . . . . . . 11 |- ((|^|A e. On /\ x e. On) -> (|^|A (_ x <-> (|^|A e. x \/ |^|A = x)))
5	3, 4	bitr3d 541	. . . . . . . . . 10 |- ((|^|A e. On /\ x e. On) -> (-. x e. |^|A <-> (|^|A e. x \/ |^|A = x)))
6		oninton 3069	. . . . . . . . . . 11 |- ((A (_ On /\ A =/= (/)) -> |^|A e. On)
7	6	adantr 398	. . . . . . . . . 10 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> |^|A e. On)
8		ssel2 2115	. . . . . . . . . . 11 |- ((A (_ On /\ x e. A) -> x e. On)
9	8	adantlr 402	. . . . . . . . . 10 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> x e. On)
10	5, 7, 9	sylanc 482	. . . . . . . . 9 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. x e. |^|A <-> (|^|A e. x \/ |^|A = x)))
11	2, 10	mpbid 202	. . . . . . . 8 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (|^|A e. x \/ |^|A = x))
12	11	ord 239	. . . . . . 7 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. |^|A e. x -> |^|A = x))
13		eqcom 1524	. . . . . . 7 |- (|^|A = x <-> x = |^|A)
14	12, 13	syl6ib 219	. . . . . 6 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. |^|A e. x -> x = |^|A))
15	14	necon1ad 1678	. . . . 5 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (x =/= |^|A -> |^|A e. x))
16	15	expimpd 382	. . . 4 |- ((A (_ On /\ A =/= (/)) -> ((x e. A /\ x =/= |^|A) -> |^|A e. x))
17		eldifsn 2516	. . . 4 |- (x e. (A \ {|^|A}) <-> (x e. A /\ x =/= |^|A))
18	16, 17	syl5ib 213	. . 3 |- ((A (_ On /\ A =/= (/)) -> (x e. (A \ {|^|A}) -> |^|A e. x))
19	18	r19.21aiv 1760	. 2 |- ((A (_ On /\ A =/= (/)) -> A.x e. (A \ {|^|A})|^|A e. x)
20		intex 2784	. . . 4 |- (A =/= (/) <-> |^|A e. V)
21		elintg 2595	. . . 4 |- (|^|A e. V -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
22	20, 21	sylbi 206	. . 3 |- (A =/= (/) -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
23	22	adantl 397	. 2 |- ((A (_ On /\ A =/= (/)) -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
24	19, 23	mpbird 203	1 |- ((A (_ On /\ A =/= (/)) -> |^|A e. |^|(A \ {|^|A}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   \/ wo 229   /\ wa 230   = wceq 997   e. wcel 999   =/= wne 1632  A.wral 1692  Vcvv 1858   \ cdif 2095   (_ wss 2098  (/)c0 2331  {csn 2461  |^|cint 2587  Oncon0 3005

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-int 2588  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009

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