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Theorem bdeqsuc 11418
Description: Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
Assertion
Ref Expression
bdeqsuc BOUNDED 𝑥 = suc 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem bdeqsuc
StepHypRef Expression
1 bdcsuc 11417 . . . 4 BOUNDED suc 𝑦
21bdss 11401 . . 3 BOUNDED 𝑥 ⊆ suc 𝑦
3 bdcv 11385 . . . . . . 7 BOUNDED 𝑥
43bdss 11401 . . . . . 6 BOUNDED 𝑦𝑥
53bdsnss 11410 . . . . . 6 BOUNDED {𝑦} ⊆ 𝑥
64, 5ax-bdan 11352 . . . . 5 BOUNDED (𝑦𝑥 ∧ {𝑦} ⊆ 𝑥)
7 unss 3172 . . . . 5 ((𝑦𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
86, 7bd0 11361 . . . 4 BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9 df-suc 4189 . . . . 5 suc 𝑦 = (𝑦 ∪ {𝑦})
109sseq1i 3048 . . . 4 (suc 𝑦𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
118, 10bd0r 11362 . . 3 BOUNDED suc 𝑦𝑥
122, 11ax-bdan 11352 . 2 BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥)
13 eqss 3038 . 2 (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥))
1412, 13bd0r 11362 1 BOUNDED 𝑥 = suc 𝑦
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  cun 2995  wss 2997  {csn 3441  suc csuc 4183  BOUNDED wbd 11349
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11350  ax-bdan 11352  ax-bdor 11353  ax-bdal 11355  ax-bdeq 11357  ax-bdel 11358  ax-bdsb 11359
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-suc 4189  df-bdc 11378
This theorem is referenced by:  bj-bdsucel  11419  bj-nn0suc0  11491
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