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Theorem bdeqsuc 14569
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 14568 . . . 4 BOUNDED suc 𝑦
21bdss 14552 . . 3 BOUNDED 𝑥 ⊆ suc 𝑦
3 bdcv 14536 . . . . . . 7 BOUNDED 𝑥
43bdss 14552 . . . . . 6 BOUNDED 𝑦𝑥
53bdsnss 14561 . . . . . 6 BOUNDED {𝑦} ⊆ 𝑥
64, 5ax-bdan 14503 . . . . 5 BOUNDED (𝑦𝑥 ∧ {𝑦} ⊆ 𝑥)
7 unss 3309 . . . . 5 ((𝑦𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
86, 7bd0 14512 . . . 4 BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9 df-suc 4371 . . . . 5 suc 𝑦 = (𝑦 ∪ {𝑦})
109sseq1i 3181 . . . 4 (suc 𝑦𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
118, 10bd0r 14513 . . 3 BOUNDED suc 𝑦𝑥
122, 11ax-bdan 14503 . 2 BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥)
13 eqss 3170 . 2 (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥))
1412, 13bd0r 14513 1 BOUNDED 𝑥 = suc 𝑦
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  cun 3127  wss 3129  {csn 3592  suc csuc 4365  BOUNDED wbd 14500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14501  ax-bdan 14503  ax-bdor 14504  ax-bdal 14506  ax-bdeq 14508  ax-bdel 14509  ax-bdsb 14510
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-suc 4371  df-bdc 14529
This theorem is referenced by:  bj-bdsucel  14570  bj-nn0suc0  14638
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