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Theorem bdeqsuc 10001
 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 10000 . . . 4 BOUNDED suc 𝑦
21bdss 9984 . . 3 BOUNDED 𝑥 ⊆ suc 𝑦
3 bdcv 9968 . . . . . . 7 BOUNDED 𝑥
43bdss 9984 . . . . . 6 BOUNDED 𝑦𝑥
53bdsnss 9993 . . . . . 6 BOUNDED {𝑦} ⊆ 𝑥
64, 5ax-bdan 9935 . . . . 5 BOUNDED (𝑦𝑥 ∧ {𝑦} ⊆ 𝑥)
7 unss 3117 . . . . 5 ((𝑦𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
86, 7bd0 9944 . . . 4 BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9 df-suc 4108 . . . . 5 suc 𝑦 = (𝑦 ∪ {𝑦})
109sseq1i 2969 . . . 4 (suc 𝑦𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
118, 10bd0r 9945 . . 3 BOUNDED suc 𝑦𝑥
122, 11ax-bdan 9935 . 2 BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥)
13 eqss 2960 . 2 (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥))
1412, 13bd0r 9945 1 BOUNDED 𝑥 = suc 𝑦
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∪ cun 2915   ⊆ wss 2917  {csn 3375  suc csuc 4102  BOUNDED wbd 9932 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdor 9936  ax-bdal 9938  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-suc 4108  df-bdc 9961 This theorem is referenced by:  bj-bdsucel  10002  bj-nn0suc0  10075
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