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Theorem bdeqsuc 12881
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 12880 . . . 4 BOUNDED suc 𝑦
21bdss 12864 . . 3 BOUNDED 𝑥 ⊆ suc 𝑦
3 bdcv 12848 . . . . . . 7 BOUNDED 𝑥
43bdss 12864 . . . . . 6 BOUNDED 𝑦𝑥
53bdsnss 12873 . . . . . 6 BOUNDED {𝑦} ⊆ 𝑥
64, 5ax-bdan 12815 . . . . 5 BOUNDED (𝑦𝑥 ∧ {𝑦} ⊆ 𝑥)
7 unss 3218 . . . . 5 ((𝑦𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
86, 7bd0 12824 . . . 4 BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9 df-suc 4261 . . . . 5 suc 𝑦 = (𝑦 ∪ {𝑦})
109sseq1i 3091 . . . 4 (suc 𝑦𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
118, 10bd0r 12825 . . 3 BOUNDED suc 𝑦𝑥
122, 11ax-bdan 12815 . 2 BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥)
13 eqss 3080 . 2 (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦𝑥))
1412, 13bd0r 12825 1 BOUNDED 𝑥 = suc 𝑦
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  cun 3037  wss 3039  {csn 3495  suc csuc 4255  BOUNDED wbd 12812
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12813  ax-bdan 12815  ax-bdor 12816  ax-bdal 12818  ax-bdeq 12820  ax-bdel 12821  ax-bdsb 12822
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-suc 4261  df-bdc 12841
This theorem is referenced by:  bj-bdsucel  12882  bj-nn0suc0  12950
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