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Theorem bdeqsuc 10367
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 10366 . . . 4 BOUNDED suc 𝑦
21bdss 10350 . . 3 BOUNDED 𝑥 ⊆ suc 𝑦
3 bdcv 10334 . . . . . . 7 BOUNDED 𝑥
43bdss 10350 . . . . . 6 BOUNDED 𝑦𝑥
53bdsnss 10359 . . . . . 6 BOUNDED {𝑦} ⊆ 𝑥
64, 5ax-bdan 10301 . . . . 5 BOUNDED (𝑦𝑥 ∧ {𝑦} ⊆ 𝑥)
7 unss 3144 . . . . 5 ((𝑦𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
86, 7bd0 10310 . . . 4 BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9 df-suc 4135 . . . . 5 suc 𝑦 = (𝑦 ∪ {𝑦})
109sseq1i 2996 . . . 4 (suc 𝑦𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
118, 10bd0r 10311 . . 3 BOUNDED suc 𝑦𝑥
122, 11ax-bdan 10301 . 2 BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥)
13 eqss 2987 . 2 (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥))
1412, 13bd0r 10311 1 BOUNDED 𝑥 = suc 𝑦
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  cun 2942  wss 2944  {csn 3402  suc csuc 4129  BOUNDED wbd 10298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-bd0 10299  ax-bdan 10301  ax-bdor 10302  ax-bdal 10304  ax-bdeq 10306  ax-bdel 10307  ax-bdsb 10308
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-suc 4135  df-bdc 10327
This theorem is referenced by:  bj-bdsucel  10368  bj-nn0suc0  10441
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