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Theorem nnsucelsuc 5985
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4183, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4199. (Contributed by Jim Kingdon, 25-Aug-2019.)
Assertion
Ref Expression
nnsucelsuc (B 𝜔 → (A B ↔ suc A suc B))

Proof of Theorem nnsucelsuc
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2083 . . . 4 (x = ∅ → (A xA ∅))
2 suceq 4088 . . . . 5 (x = ∅ → suc x = suc ∅)
32eleq2d 2089 . . . 4 (x = ∅ → (suc A suc x ↔ suc A suc ∅))
41, 3imbi12d 223 . . 3 (x = ∅ → ((A x → suc A suc x) ↔ (A ∅ → suc A suc ∅)))
5 eleq2 2083 . . . 4 (x = y → (A xA y))
6 suceq 4088 . . . . 5 (x = y → suc x = suc y)
76eleq2d 2089 . . . 4 (x = y → (suc A suc x ↔ suc A suc y))
85, 7imbi12d 223 . . 3 (x = y → ((A x → suc A suc x) ↔ (A y → suc A suc y)))
9 eleq2 2083 . . . 4 (x = suc y → (A xA suc y))
10 suceq 4088 . . . . 5 (x = suc y → suc x = suc suc y)
1110eleq2d 2089 . . . 4 (x = suc y → (suc A suc x ↔ suc A suc suc y))
129, 11imbi12d 223 . . 3 (x = suc y → ((A x → suc A suc x) ↔ (A suc y → suc A suc suc y)))
13 eleq2 2083 . . . 4 (x = B → (A xA B))
14 suceq 4088 . . . . 5 (x = B → suc x = suc B)
1514eleq2d 2089 . . . 4 (x = B → (suc A suc x ↔ suc A suc B))
1613, 15imbi12d 223 . . 3 (x = B → ((A x → suc A suc x) ↔ (A B → suc A suc B)))
17 noel 3205 . . . 4 ¬ A
1817pm2.21i 562 . . 3 (A ∅ → suc A suc ∅)
19 elsuci 4089 . . . . . . . 8 (A suc y → (A y A = y))
2019adantl 262 . . . . . . 7 (((A y → suc A suc y) A suc y) → (A y A = y))
21 ax-ia1 99 . . . . . . . 8 (((A y → suc A suc y) A suc y) → (A y → suc A suc y))
22 suceq 4088 . . . . . . . . 9 (A = y → suc A = suc y)
2322a1i 9 . . . . . . . 8 (((A y → suc A suc y) A suc y) → (A = y → suc A = suc y))
2421, 23orim12d 687 . . . . . . 7 (((A y → suc A suc y) A suc y) → ((A y A = y) → (suc A suc y suc A = suc y)))
2520, 24mpd 13 . . . . . 6 (((A y → suc A suc y) A suc y) → (suc A suc y suc A = suc y))
26 vex 2538 . . . . . . . 8 y V
2726sucex 4175 . . . . . . 7 suc y V
2827elsuc2 4093 . . . . . 6 (suc A suc suc y ↔ (suc A suc y suc A = suc y))
2925, 28sylibr 137 . . . . 5 (((A y → suc A suc y) A suc y) → suc A suc suc y)
3029ex 108 . . . 4 ((A y → suc A suc y) → (A suc y → suc A suc suc y))
3130a1i 9 . . 3 (y 𝜔 → ((A y → suc A suc y) → (A suc y → suc A suc suc y)))
324, 8, 12, 16, 18, 31finds 4250 . 2 (B 𝜔 → (A B → suc A suc B))
33 nnon 4259 . . 3 (B 𝜔 → B On)
34 onsucelsucr 4183 . . 3 (B On → (suc A suc BA B))
3533, 34syl 14 . 2 (B 𝜔 → (suc A suc BA B))
3632, 35impbid 120 1 (B 𝜔 → (A B ↔ suc A suc B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 616   = wceq 1228   wcel 1374  c0 3201  Oncon0 4049  suc csuc 4051  𝜔com 4240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241
This theorem is referenced by:  nnsucsssuc  5986  nntri3or  5987  nnaordi  5992
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