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Theorem nmembers1lem3 6270
 Description: Lemma for nmembers1 6271. If the interval from one to a natural is in a given natural, extending it by one puts it in the next natural. (Contributed by Scott Fenton, 3-Aug-2019.)
Assertion
Ref Expression
nmembers1lem3 ((A Nn B Nn ) → ({m Nn (1cc m mc A)} B → {m Nn (1cc m mc (A +c 1c))} (B +c 1c)))
Distinct variable groups:   A,m   B,m

Proof of Theorem nmembers1lem3
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnltp1c 6262 . . . . . . . . . . . 12 (A NnA <c (A +c 1c))
2 nnnc 6146 . . . . . . . . . . . . 13 (A NnA NC )
3 peano2 4403 . . . . . . . . . . . . . 14 (A Nn → (A +c 1c) Nn )
4 nnnc 6146 . . . . . . . . . . . . . 14 ((A +c 1c) Nn → (A +c 1c) NC )
53, 4syl 15 . . . . . . . . . . . . 13 (A Nn → (A +c 1c) NC )
6 ltlenlec 6207 . . . . . . . . . . . . 13 ((A NC (A +c 1c) NC ) → (A <c (A +c 1c) ↔ (Ac (A +c 1c) ¬ (A +c 1c) ≤c A)))
72, 5, 6syl2anc 642 . . . . . . . . . . . 12 (A Nn → (A <c (A +c 1c) ↔ (Ac (A +c 1c) ¬ (A +c 1c) ≤c A)))
81, 7mpbid 201 . . . . . . . . . . 11 (A Nn → (Ac (A +c 1c) ¬ (A +c 1c) ≤c A))
98simprd 449 . . . . . . . . . 10 (A Nn → ¬ (A +c 1c) ≤c A)
109adantr 451 . . . . . . . . 9 ((A Nn B Nn ) → ¬ (A +c 1c) ≤c A)
1110intnand 882 . . . . . . . 8 ((A Nn B Nn ) → ¬ (1cc (A +c 1c) (A +c 1c) ≤c A))
1211a1d 22 . . . . . . 7 ((A Nn B Nn ) → ((A +c 1c) Nn → ¬ (1cc (A +c 1c) (A +c 1c) ≤c A)))
13 breq2 4643 . . . . . . . . . . 11 (m = (A +c 1c) → (1cc m ↔ 1cc (A +c 1c)))
14 breq1 4642 . . . . . . . . . . 11 (m = (A +c 1c) → (mc A ↔ (A +c 1c) ≤c A))
1513, 14anbi12d 691 . . . . . . . . . 10 (m = (A +c 1c) → ((1cc m mc A) ↔ (1cc (A +c 1c) (A +c 1c) ≤c A)))
1615elrab 2994 . . . . . . . . 9 ((A +c 1c) {m Nn (1cc m mc A)} ↔ ((A +c 1c) Nn (1cc (A +c 1c) (A +c 1c) ≤c A)))
1716notbii 287 . . . . . . . 8 (¬ (A +c 1c) {m Nn (1cc m mc A)} ↔ ¬ ((A +c 1c) Nn (1cc (A +c 1c) (A +c 1c) ≤c A)))
18 imnan 411 . . . . . . . 8 (((A +c 1c) Nn → ¬ (1cc (A +c 1c) (A +c 1c) ≤c A)) ↔ ¬ ((A +c 1c) Nn (1cc (A +c 1c) (A +c 1c) ≤c A)))
1917, 18bitr4i 243 . . . . . . 7 (¬ (A +c 1c) {m Nn (1cc m mc A)} ↔ ((A +c 1c) Nn → ¬ (1cc (A +c 1c) (A +c 1c) ≤c A)))
2012, 19sylibr 203 . . . . . 6 ((A Nn B Nn ) → ¬ (A +c 1c) {m Nn (1cc m mc A)})
213adantr 451 . . . . . . 7 ((A Nn B Nn ) → (A +c 1c) Nn )
22 elcomplg 3218 . . . . . . 7 ((A +c 1c) Nn → ((A +c 1c) ∼ {m Nn (1cc m mc A)} ↔ ¬ (A +c 1c) {m Nn (1cc m mc A)}))
2321, 22syl 15 . . . . . 6 ((A Nn B Nn ) → ((A +c 1c) ∼ {m Nn (1cc m mc A)} ↔ ¬ (A +c 1c) {m Nn (1cc m mc A)}))
2420, 23mpbird 223 . . . . 5 ((A Nn B Nn ) → (A +c 1c) ∼ {m Nn (1cc m mc A)})
25 nnnc 6146 . . . . . . . . . . . . . 14 (x Nnx NC )
26 ncslesuc 6267 . . . . . . . . . . . . . 14 ((x NC A NC ) → (xc (A +c 1c) ↔ (xc A x = (A +c 1c))))
2725, 2, 26syl2an 463 . . . . . . . . . . . . 13 ((x Nn A Nn ) → (xc (A +c 1c) ↔ (xc A x = (A +c 1c))))
2827expcom 424 . . . . . . . . . . . 12 (A Nn → (x Nn → (xc (A +c 1c) ↔ (xc A x = (A +c 1c)))))
2928adantrd 454 . . . . . . . . . . 11 (A Nn → ((x Nn 1cc x) → (xc (A +c 1c) ↔ (xc A x = (A +c 1c)))))
3029adantr 451 . . . . . . . . . 10 ((A Nn B Nn ) → ((x Nn 1cc x) → (xc (A +c 1c) ↔ (xc A x = (A +c 1c)))))
3130pm5.32d 620 . . . . . . . . 9 ((A Nn B Nn ) → (((x Nn 1cc x) xc (A +c 1c)) ↔ ((x Nn 1cc x) (xc A x = (A +c 1c)))))
32 anass 630 . . . . . . . . 9 (((x Nn 1cc x) xc (A +c 1c)) ↔ (x Nn (1cc x xc (A +c 1c))))
33 andi 837 . . . . . . . . . 10 (((x Nn 1cc x) (xc A x = (A +c 1c))) ↔ (((x Nn 1cc x) xc A) ((x Nn 1cc x) x = (A +c 1c))))
34 anass 630 . . . . . . . . . . 11 (((x Nn 1cc x) xc A) ↔ (x Nn (1cc x xc A)))
3534orbi1i 506 . . . . . . . . . 10 ((((x Nn 1cc x) xc A) ((x Nn 1cc x) x = (A +c 1c))) ↔ ((x Nn (1cc x xc A)) ((x Nn 1cc x) x = (A +c 1c))))
3633, 35bitri 240 . . . . . . . . 9 (((x Nn 1cc x) (xc A x = (A +c 1c))) ↔ ((x Nn (1cc x xc A)) ((x Nn 1cc x) x = (A +c 1c))))
3731, 32, 363bitr3g 278 . . . . . . . 8 ((A Nn B Nn ) → ((x Nn (1cc x xc (A +c 1c))) ↔ ((x Nn (1cc x xc A)) ((x Nn 1cc x) x = (A +c 1c)))))
38 1cnc 6139 . . . . . . . . . . . . . . . 16 1c NC
39 addlecncs 6209 . . . . . . . . . . . . . . . 16 ((1c NC A NC ) → 1cc (1c +c A))
4038, 2, 39sylancr 644 . . . . . . . . . . . . . . 15 (A Nn → 1cc (1c +c A))
41 addccom 4406 . . . . . . . . . . . . . . 15 (A +c 1c) = (1c +c A)
4240, 41syl6breqr 4679 . . . . . . . . . . . . . 14 (A Nn → 1cc (A +c 1c))
433, 42jca 518 . . . . . . . . . . . . 13 (A Nn → ((A +c 1c) Nn 1cc (A +c 1c)))
44 eleq1 2413 . . . . . . . . . . . . . 14 (x = (A +c 1c) → (x Nn ↔ (A +c 1c) Nn ))
45 breq2 4643 . . . . . . . . . . . . . 14 (x = (A +c 1c) → (1cc x ↔ 1cc (A +c 1c)))
4644, 45anbi12d 691 . . . . . . . . . . . . 13 (x = (A +c 1c) → ((x Nn 1cc x) ↔ ((A +c 1c) Nn 1cc (A +c 1c))))
4743, 46syl5ibrcom 213 . . . . . . . . . . . 12 (A Nn → (x = (A +c 1c) → (x Nn 1cc x)))
4847adantr 451 . . . . . . . . . . 11 ((A Nn B Nn ) → (x = (A +c 1c) → (x Nn 1cc x)))
4948pm4.71rd 616 . . . . . . . . . 10 ((A Nn B Nn ) → (x = (A +c 1c) ↔ ((x Nn 1cc x) x = (A +c 1c))))
5049bicomd 192 . . . . . . . . 9 ((A Nn B Nn ) → (((x Nn 1cc x) x = (A +c 1c)) ↔ x = (A +c 1c)))
5150orbi2d 682 . . . . . . . 8 ((A Nn B Nn ) → (((x Nn (1cc x xc A)) ((x Nn 1cc x) x = (A +c 1c))) ↔ ((x Nn (1cc x xc A)) x = (A +c 1c))))
5237, 51bitrd 244 . . . . . . 7 ((A Nn B Nn ) → ((x Nn (1cc x xc (A +c 1c))) ↔ ((x Nn (1cc x xc A)) x = (A +c 1c))))
53 breq2 4643 . . . . . . . . 9 (m = x → (1cc m ↔ 1cc x))
54 breq1 4642 . . . . . . . . 9 (m = x → (mc (A +c 1c) ↔ xc (A +c 1c)))
5553, 54anbi12d 691 . . . . . . . 8 (m = x → ((1cc m mc (A +c 1c)) ↔ (1cc x xc (A +c 1c))))
5655elrab 2994 . . . . . . 7 (x {m Nn (1cc m mc (A +c 1c))} ↔ (x Nn (1cc x xc (A +c 1c))))
57 elun 3220 . . . . . . . 8 (x ({m Nn (1cc m mc A)} ∪ {(A +c 1c)}) ↔ (x {m Nn (1cc m mc A)} x {(A +c 1c)}))
58 breq1 4642 . . . . . . . . . . 11 (m = x → (mc Axc A))
5953, 58anbi12d 691 . . . . . . . . . 10 (m = x → ((1cc m mc A) ↔ (1cc x xc A)))
6059elrab 2994 . . . . . . . . 9 (x {m Nn (1cc m mc A)} ↔ (x Nn (1cc x xc A)))
61 elsn 3748 . . . . . . . . 9 (x {(A +c 1c)} ↔ x = (A +c 1c))
6260, 61orbi12i 507 . . . . . . . 8 ((x {m Nn (1cc m mc A)} x {(A +c 1c)}) ↔ ((x Nn (1cc x xc A)) x = (A +c 1c)))
6357, 62bitri 240 . . . . . . 7 (x ({m Nn (1cc m mc A)} ∪ {(A +c 1c)}) ↔ ((x Nn (1cc x xc A)) x = (A +c 1c)))
6452, 56, 633bitr4g 279 . . . . . 6 ((A Nn B Nn ) → (x {m Nn (1cc m mc (A +c 1c))} ↔ x ({m Nn (1cc m mc A)} ∪ {(A +c 1c)})))
6564eqrdv 2351 . . . . 5 ((A Nn B Nn ) → {m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {(A +c 1c)}))
66 sneq 3744 . . . . . . . 8 (y = (A +c 1c) → {y} = {(A +c 1c)})
6766uneq2d 3418 . . . . . . 7 (y = (A +c 1c) → ({m Nn (1cc m mc A)} ∪ {y}) = ({m Nn (1cc m mc A)} ∪ {(A +c 1c)}))
6867eqeq2d 2364 . . . . . 6 (y = (A +c 1c) → ({m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {y}) ↔ {m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {(A +c 1c)})))
6968rspcev 2955 . . . . 5 (((A +c 1c) ∼ {m Nn (1cc m mc A)} {m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {(A +c 1c)})) → y ∼ {m Nn (1cc m mc A)} {m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {y}))
7024, 65, 69syl2anc 642 . . . 4 ((A Nn B Nn ) → y ∼ {m Nn (1cc m mc A)} {m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {y}))
71 compleq 3243 . . . . . 6 (x = {m Nn (1cc m mc A)} → ∼ x = ∼ {m Nn (1cc m mc A)})
72 uneq1 3411 . . . . . . 7 (x = {m Nn (1cc m mc A)} → (x ∪ {y}) = ({m Nn (1cc m mc A)} ∪ {y}))
7372eqeq2d 2364 . . . . . 6 (x = {m Nn (1cc m mc A)} → ({m Nn (1cc m mc (A +c 1c))} = (x ∪ {y}) ↔ {m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {y})))
7471, 73rexeqbidv 2820 . . . . 5 (x = {m Nn (1cc m mc A)} → (y x{m Nn (1cc m mc (A +c 1c))} = (x ∪ {y}) ↔ y ∼ {m Nn (1cc m mc A)} {m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {y})))
7574rspcev 2955 . . . 4 (({m Nn (1cc m mc A)} B y ∼ {m Nn (1cc m mc A)} {m Nn (1cc m mc (A +c 1c))} = ({m Nn (1cc m mc A)} ∪ {y})) → x B y x{m Nn (1cc m mc (A +c 1c))} = (x ∪ {y}))
7670, 75sylan2 460 . . 3 (({m Nn (1cc m mc A)} B (A Nn B Nn )) → x B y x{m Nn (1cc m mc (A +c 1c))} = (x ∪ {y}))
77 elsuc 4413 . . 3 ({m Nn (1cc m mc (A +c 1c))} (B +c 1c) ↔ x B y x{m Nn (1cc m mc (A +c 1c))} = (x ∪ {y}))
7876, 77sylibr 203 . 2 (({m Nn (1cc m mc A)} B (A Nn B Nn )) → {m Nn (1cc m mc (A +c 1c))} (B +c 1c))
7978expcom 424 1 ((A Nn B Nn ) → ({m Nn (1cc m mc A)} B → {m Nn (1cc m mc (A +c 1c))} (B +c 1c)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {crab 2618   ∼ ccompl 3205   ∪ cun 3207  {csn 3737  1cc1c 4134   Nn cnnc 4373   +c cplc 4375   class class class wbr 4639   NC cncs 6088   ≤c clec 6089
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