ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnsucuniel GIF version

Theorem nnsucuniel 6498
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 4511). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4532). (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 3428 . . . . . . 7 ¬ 𝐴 ∈ ∅
2 uni0 3838 . . . . . . . 8 ∅ = ∅
32eleq2i 2244 . . . . . . 7 (𝐴 ∅ ↔ 𝐴 ∈ ∅)
41, 3mtbir 671 . . . . . 6 ¬ 𝐴
5 unieq 3820 . . . . . . 7 (𝐵 = ∅ → 𝐵 = ∅)
65eleq2d 2247 . . . . . 6 (𝐵 = ∅ → (𝐴 𝐵𝐴 ∅))
74, 6mtbiri 675 . . . . 5 (𝐵 = ∅ → ¬ 𝐴 𝐵)
87pm2.21d 619 . . . 4 (𝐵 = ∅ → (𝐴 𝐵 → suc 𝐴𝐵))
98adantl 277 . . 3 ((𝐵 ∈ ω ∧ 𝐵 = ∅) → (𝐴 𝐵 → suc 𝐴𝐵))
10 unieq 3820 . . . . . . . . . . . 12 (𝐵 = suc 𝑛 𝐵 = suc 𝑛)
1110eleq2d 2247 . . . . . . . . . . 11 (𝐵 = suc 𝑛 → (𝐴 𝐵𝐴 suc 𝑛))
1211ad2antll 491 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 𝐵𝐴 suc 𝑛))
1312biimpa 296 . . . . . . . . 9 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐴 suc 𝑛)
14 simplrl 535 . . . . . . . . . . 11 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝑛 ∈ ω)
15 nnord 4613 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
16 ordtr 4380 . . . . . . . . . . . . 13 (Ord 𝑛 → Tr 𝑛)
1715, 16syl 14 . . . . . . . . . . . 12 (𝑛 ∈ ω → Tr 𝑛)
18 vex 2742 . . . . . . . . . . . . 13 𝑛 ∈ V
1918unisuc 4415 . . . . . . . . . . . 12 (Tr 𝑛 suc 𝑛 = 𝑛)
2017, 19sylib 122 . . . . . . . . . . 11 (𝑛 ∈ ω → suc 𝑛 = 𝑛)
2114, 20syl 14 . . . . . . . . . 10 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝑛 = 𝑛)
2221eleq2d 2247 . . . . . . . . 9 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → (𝐴 suc 𝑛𝐴𝑛))
2313, 22mpbid 147 . . . . . . . 8 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐴𝑛)
24 nnsucelsuc 6494 . . . . . . . . 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 147 . . . . . . 7 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝐴 ∈ suc 𝑛)
27 simplrr 536 . . . . . . 7 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐵 = suc 𝑛)
2826, 27eleqtrrd 2257 . . . . . 6 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝐴𝐵)
2928ex 115 . . . . 5 ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 𝐵 → suc 𝐴𝐵))
3029rexlimdvaa 2595 . . . 4 (𝐵 ∈ ω → (∃𝑛 ∈ ω 𝐵 = suc 𝑛 → (𝐴 𝐵 → suc 𝐴𝐵)))
3130imp 124 . . 3 ((𝐵 ∈ ω ∧ ∃𝑛 ∈ ω 𝐵 = suc 𝑛) → (𝐴 𝐵 → suc 𝐴𝐵))
32 nn0suc 4605 . . 3 (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∃𝑛 ∈ ω 𝐵 = suc 𝑛))
339, 31, 32mpjaodan 798 . 2 (𝐵 ∈ ω → (𝐴 𝐵 → suc 𝐴𝐵))
34 sucunielr 4511 . 2 (suc 𝐴𝐵𝐴 𝐵)
3533, 34impbid1 142 1 (𝐵 ∈ ω → (𝐴 𝐵 ↔ suc 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wrex 2456  c0 3424   cuni 3811  Tr wtr 4103  Ord word 4364  suc csuc 4367  ωcom 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator