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Theorem nnsucuniel 6359
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 4396). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4416). (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 3337 . . . . . . 7 ¬ 𝐴 ∈ ∅
2 uni0 3733 . . . . . . . 8 ∅ = ∅
32eleq2i 2184 . . . . . . 7 (𝐴 ∅ ↔ 𝐴 ∈ ∅)
41, 3mtbir 645 . . . . . 6 ¬ 𝐴
5 unieq 3715 . . . . . . 7 (𝐵 = ∅ → 𝐵 = ∅)
65eleq2d 2187 . . . . . 6 (𝐵 = ∅ → (𝐴 𝐵𝐴 ∅))
74, 6mtbiri 649 . . . . 5 (𝐵 = ∅ → ¬ 𝐴 𝐵)
87pm2.21d 593 . . . 4 (𝐵 = ∅ → (𝐴 𝐵 → suc 𝐴𝐵))
98adantl 275 . . 3 ((𝐵 ∈ ω ∧ 𝐵 = ∅) → (𝐴 𝐵 → suc 𝐴𝐵))
10 unieq 3715 . . . . . . . . . . . 12 (𝐵 = suc 𝑛 𝐵 = suc 𝑛)
1110eleq2d 2187 . . . . . . . . . . 11 (𝐵 = suc 𝑛 → (𝐴 𝐵𝐴 suc 𝑛))
1211ad2antll 482 . . . . . . . . . 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 509 . . . . . . . . . . 11 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝑛 ∈ ω)
15 nnord 4495 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
16 ordtr 4270 . . . . . . . . . . . . 13 (Ord 𝑛 → Tr 𝑛)
1715, 16syl 14 . . . . . . . . . . . 12 (𝑛 ∈ ω → Tr 𝑛)
18 vex 2663 . . . . . . . . . . . . 13 𝑛 ∈ V
1918unisuc 4305 . . . . . . . . . . . 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 2187 . . . . . . . . 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 6355 . . . . . . . . 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 510 . . . . . . 7 (((𝐵 ∈ ω ∧ (𝑛 ∈ ω ∧ 𝐵 = suc 𝑛)) ∧ 𝐴 𝐵) → 𝐵 = suc 𝑛)
2826, 27eleqtrrd 2197 . . . . . 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 2527 . . . 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 4488 . . 3 (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∃𝑛 ∈ ω 𝐵 = suc 𝑛))
339, 31, 32mpjaodan 772 . 2 (𝐵 ∈ ω → (𝐴 𝐵 → suc 𝐴𝐵))
34 sucunielr 4396 . 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 1316  wcel 1465  wrex 2394  c0 3333   cuni 3706  Tr wtr 3996  Ord word 4254  suc csuc 4257  ωcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475
This theorem is referenced by: (None)
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