Theorem onmindif2 3067

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 3021	. . . . . . . . . 10 |- ((A (_ On /\ x e. A) -> -. x e. |^|A)
2	1	adantlr 395	. . . . . . . . 9 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> -. x e. |^|A)
3		ontri1 2987	. . . . . . . . . . 11 |- ((|^|A e. On /\ x e. On) -> (|^|A (_ x <-> -. x e. |^|A))
4		onsseleq 3005	. . . . . . . . . . 11 |- ((|^|A e. On /\ x e. On) -> (|^|A (_ x <-> (|^|A e. x \/ |^|A = x)))
5	3, 4	bitr3d 532	. . . . . . . . . 10 |- ((|^|A e. On /\ x e. On) -> (-. x e. |^|A <-> (|^|A e. x \/ |^|A = x)))
6		oninton 3018	. . . . . . . . . . 11 |- ((A (_ On /\ A =/= (/)) -> |^|A e. On)
7	6	adantr 391	. . . . . . . . . 10 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> |^|A e. On)
8		ssel2 2067	. . . . . . . . . . 11 |- ((A (_ On /\ x e. A) -> x e. On)
9	8	adantlr 395	. . . . . . . . . 10 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> x e. On)
10	5, 7, 9	sylanc 473	. . . . . . . . 9 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. x e. |^|A <-> (|^|A e. x \/ |^|A = x)))
11	2, 10	mpbid 195	. . . . . . . 8 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (|^|A e. x \/ |^|A = x))
12	11	ord 232	. . . . . . 7 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. |^|A e. x -> |^|A = x))
13		eqcom 1480	. . . . . . 7 |- (|^|A = x <-> x = |^|A)
14	12, 13	syl6ib 212	. . . . . 6 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (-. |^|A e. x -> x = |^|A))
15	14	necon1ad 1634	. . . . 5 |- (((A (_ On /\ A =/= (/)) /\ x e. A) -> (x =/= |^|A -> |^|A e. x))
16	15	expimpd 375	. . . 4 |- ((A (_ On /\ A =/= (/)) -> ((x e. A /\ x =/= |^|A) -> |^|A e. x))
17		eldifsn 2466	. . . 4 |- (x e. (A \ {|^|A}) <-> (x e. A /\ x =/= |^|A))
18	16, 17	syl5ib 206	. . 3 |- ((A (_ On /\ A =/= (/)) -> (x e. (A \ {|^|A}) -> |^|A e. x))
19	18	r19.21aiv 1716	. 2 |- ((A (_ On /\ A =/= (/)) -> A.x e. (A \ {|^|A})|^|A e. x)
20		intex 2734	. . . 4 |- (A =/= (/) <-> |^|A e. V)
21		elintg 2545	. . . 4 |- (|^|A e. V -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
22	20, 21	sylbi 199	. . 3 |- (A =/= (/) -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
23	22	adantl 390	. 2 |- ((A (_ On /\ A =/= (/)) -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
24	19, 23	mpbird 196	1 |- ((A (_ On /\ A =/= (/)) -> |^|A e. |^|(A \ {|^|A}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  Vcvv 1814   \ cdif 2047   (_ wss 2050  (/)c0 2283  {csn 2413  |^|cint 2537  Oncon0 2954

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-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-sep 2708  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-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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