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Theorem nnsucuniel 6395
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 4430). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4450). (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 3368 . . . . . . 7 ¬ 𝐴 ∈ ∅
2 uni0 3767 . . . . . . . 8 ∅ = ∅
32eleq2i 2207 . . . . . . 7 (𝐴 ∅ ↔ 𝐴 ∈ ∅)
41, 3mtbir 661 . . . . . 6 ¬ 𝐴
5 unieq 3749 . . . . . . 7 (𝐵 = ∅ → 𝐵 = ∅)
65eleq2d 2210 . . . . . 6 (𝐵 = ∅ → (𝐴 𝐵𝐴 ∅))
74, 6mtbiri 665 . . . . 5 (𝐵 = ∅ → ¬ 𝐴 𝐵)
87pm2.21d 609 . . . 4 (𝐵 = ∅ → (𝐴 𝐵 → suc 𝐴𝐵))
98adantl 275 . . 3 ((𝐵 ∈ ω ∧ 𝐵 = ∅) → (𝐴 𝐵 → suc 𝐴𝐵))
10 unieq 3749 . . . . . . . . . . . 12 (𝐵 = suc 𝑛 𝐵 = suc 𝑛)
1110eleq2d 2210 . . . . . . . . . . 11 (𝐵 = suc 𝑛 → (𝐴 𝐵𝐴 suc 𝑛))
1211ad2antll 483 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 𝐵𝐴 suc 𝑛))
1312biimpa 294 . . . . . . . . 9 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐴 suc 𝑛)
14 simplrl 525 . . . . . . . . . . 11 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝑛 ∈ ω)
15 nnord 4529 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
16 ordtr 4304 . . . . . . . . . . . . 13 (Ord 𝑛 → Tr 𝑛)
1715, 16syl 14 . . . . . . . . . . . 12 (𝑛 ∈ ω → Tr 𝑛)
18 vex 2690 . . . . . . . . . . . . 13 𝑛 ∈ V
1918unisuc 4339 . . . . . . . . . . . 12 (Tr 𝑛 suc 𝑛 = 𝑛)
2017, 19sylib 121 . . . . . . . . . . 11 (𝑛 ∈ ω → suc 𝑛 = 𝑛)
2114, 20syl 14 . . . . . . . . . 10 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝑛 = 𝑛)
2221eleq2d 2210 . . . . . . . . 9 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → (𝐴 suc 𝑛𝐴𝑛))
2313, 22mpbid 146 . . . . . . . 8 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐴𝑛)
24 nnsucelsuc 6391 . . . . . . . . 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 146 . . . . . . 7 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝐴 ∈ suc 𝑛)
27 simplrr 526 . . . . . . 7 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐵 = suc 𝑛)
2826, 27eleqtrrd 2220 . . . . . 6 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → suc 𝐴𝐵)
2928ex 114 . . . . 5 ((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) → (𝐴 𝐵 → suc 𝐴𝐵))
3029rexlimdvaa 2551 . . . 4 (𝐵 ∈ ω → (∃𝑛 ∈ ω 𝐵 = suc 𝑛 → (𝐴 𝐵 → suc 𝐴𝐵)))
3130imp 123 . . 3 ((𝐵 ∈ ω ∧ ∃𝑛 ∈ ω 𝐵 = suc 𝑛) → (𝐴 𝐵 → suc 𝐴𝐵))
32 nn0suc 4522 . . 3 (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∃𝑛 ∈ ω 𝐵 = suc 𝑛))
339, 31, 32mpjaodan 788 . 2 (𝐵 ∈ ω → (𝐴 𝐵 → suc 𝐴𝐵))
34 sucunielr 4430 . 2 (suc 𝐴𝐵𝐴 𝐵)
3533, 34impbid1 141 1 (𝐵 ∈ ω → (𝐴 𝐵 ↔ suc 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  wrex 2418  c0 3364   cuni 3740  Tr wtr 4030  Ord word 4288  suc csuc 4291  ωcom 4508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-uni 3741  df-int 3776  df-tr 4031  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509
This theorem is referenced by: (None)
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