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Theorem nnsucuniel 6159
 Description: Given an element 𝐴 of the union of a natural number 𝐵, suc 𝐴 is an element of 𝐵 itself. The reverse direction holds for all ordinals (sucunielr 4282). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4302). (Contributed by Jim Kingdon, 13-Mar-2022.)
Assertion
Ref Expression
nnsucuniel (𝐵 ∈ ω → (𝐴 𝐵 ↔ suc 𝐴𝐵))

Proof of Theorem nnsucuniel
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 noel 3271 . . . . . . 7 ¬ 𝐴 ∈ ∅
2 uni0 3648 . . . . . . . 8 ∅ = ∅
32eleq2i 2149 . . . . . . 7 (𝐴 ∅ ↔ 𝐴 ∈ ∅)
41, 3mtbir 629 . . . . . 6 ¬ 𝐴
5 unieq 3630 . . . . . . 7 (𝐵 = ∅ → 𝐵 = ∅)
65eleq2d 2152 . . . . . 6 (𝐵 = ∅ → (𝐴 𝐵𝐴 ∅))
74, 6mtbiri 633 . . . . 5 (𝐵 = ∅ → ¬ 𝐴 𝐵)
87pm2.21d 582 . . . 4 (𝐵 = ∅ → (𝐴 𝐵 → suc 𝐴𝐵))
98adantl 271 . . 3 ((𝐵 ∈ ω ∧ 𝐵 = ∅) → (𝐴 𝐵 → suc 𝐴𝐵))
10 unieq 3630 . . . . . . . . . . . 12 (𝐵 = suc 𝑛 𝐵 = suc 𝑛)
1110eleq2d 2152 . . . . . . . . . . 11 (𝐵 = suc 𝑛 → (𝐴 𝐵𝐴 suc 𝑛))
1211ad2antll 475 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 𝐵𝐴 suc 𝑛))
1312biimpa 290 . . . . . . . . 9 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐴 suc 𝑛)
14 simplrl 502 . . . . . . . . . . 11 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝑛 ∈ ω)
15 nnord 4380 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
16 ordtr 4161 . . . . . . . . . . . . 13 (Ord 𝑛 → Tr 𝑛)
1715, 16syl 14 . . . . . . . . . . . 12 (𝑛 ∈ ω → Tr 𝑛)
18 vex 2613 . . . . . . . . . . . . 13 𝑛 ∈ V
1918unisuc 4196 . . . . . . . . . . . 12 (Tr 𝑛 suc 𝑛 = 𝑛)
2017, 19sylib 120 . . . . . . . . . . 11 (𝑛 ∈ ω → suc 𝑛 = 𝑛)
2114, 20syl 14 . . . . . . . . . 10 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝑛 = 𝑛)
2221eleq2d 2152 . . . . . . . . 9 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → (𝐴 suc 𝑛𝐴𝑛))
2313, 22mpbid 145 . . . . . . . 8 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐴𝑛)
24 nnsucelsuc 6155 . . . . . . . . 9 (𝑛 ∈ ω → (𝐴𝑛 ↔ suc 𝐴 ∈ suc 𝑛))
2514, 24syl 14 . . . . . . . 8 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → (𝐴𝑛 ↔ suc 𝐴 ∈ suc 𝑛))
2623, 25mpbid 145 . . . . . . 7 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝐴 ∈ suc 𝑛)
27 simplrr 503 . . . . . . 7 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐵 = suc 𝑛)
2826, 27eleqtrrd 2162 . . . . . 6 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝐴𝐵)
2928ex 113 . . . . 5 ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 𝐵 → suc 𝐴𝐵))
3029rexlimdvaa 2483 . . . 4 (𝐵 ∈ ω → (∃𝑛 ∈ ω 𝐵 = suc 𝑛 → (𝐴 𝐵 → suc 𝐴𝐵)))
3130imp 122 . . 3 ((𝐵 ∈ ω ∧ ∃𝑛 ∈ ω 𝐵 = suc 𝑛) → (𝐴 𝐵 → suc 𝐴𝐵))
32 nn0suc 4373 . . 3 (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∃𝑛 ∈ ω 𝐵 = suc 𝑛))
339, 31, 32mpjaodan 745 . 2 (𝐵 ∈ ω → (𝐴 𝐵 → suc 𝐴𝐵))
34 sucunielr 4282 . 2 (suc 𝐴𝐵𝐴 𝐵)
3533, 34impbid1 140 1 (𝐵 ∈ ω → (𝐴 𝐵 ↔ suc 𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285   ∈ wcel 1434  ∃wrex 2354  ∅c0 3267  ∪ cuni 3621  Tr wtr 3895  Ord word 4145  suc csuc 4148  ωcom 4359 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-iinf 4357 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-int 3657  df-tr 3896  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360 This theorem is referenced by: (None)
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