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Theorem bdeqsuc 9316
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 9315 . . . 4 BOUNDED suc y
21bdss 9299 . . 3 BOUNDED x ⊆ suc y
3 bdcv 9283 . . . . . . 7 BOUNDED x
43bdss 9299 . . . . . 6 BOUNDED yx
53bdsnss 9308 . . . . . 6 BOUNDED {y} ⊆ x
64, 5ax-bdan 9250 . . . . 5 BOUNDED (yx {y} ⊆ x)
7 unss 3111 . . . . 5 ((yx {y} ⊆ x) ↔ (y ∪ {y}) ⊆ x)
86, 7bd0 9259 . . . 4 BOUNDED (y ∪ {y}) ⊆ x
9 df-suc 4074 . . . . 5 suc y = (y ∪ {y})
109sseq1i 2963 . . . 4 (suc yx ↔ (y ∪ {y}) ⊆ x)
118, 10bd0r 9260 . . 3 BOUNDED suc yx
122, 11ax-bdan 9250 . 2 BOUNDED (x ⊆ suc y suc yx)
13 eqss 2954 . 2 (x = suc y ↔ (x ⊆ suc y suc yx))
1412, 13bd0r 9260 1 BOUNDED x = suc y
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  cun 2909  wss 2911  {csn 3367  suc csuc 4068  BOUNDED wbd 9247
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9248  ax-bdan 9250  ax-bdor 9251  ax-bdal 9253  ax-bdeq 9255  ax-bdel 9256  ax-bdsb 9257
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-suc 4074  df-bdc 9276
This theorem is referenced by:  bj-bdsucel  9317  bj-nn0suc0  9384
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