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Theorem bj-inf2vn 9428
 Description: A sufficient condition for 𝜔 to be a set. See bj-inf2vn2 9429 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-inf2vn.1 BOUNDED A
Assertion
Ref Expression
bj-inf2vn (A 𝑉 → (x(x A ↔ (x = ∅ y A x = suc y)) → A = 𝜔))
Distinct variable group:   x,y,A
Allowed substitution hints:   𝑉(x,y)

Proof of Theorem bj-inf2vn
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem1 9424 . . 3 (x(x A ↔ (x = ∅ y A x = suc y)) → Ind A)
2 bi1 111 . . . . . . 7 ((x A ↔ (x = ∅ y A x = suc y)) → (x A → (x = ∅ y A x = suc y)))
32alimi 1341 . . . . . 6 (x(x A ↔ (x = ∅ y A x = suc y)) → x(x A → (x = ∅ y A x = suc y)))
4 df-ral 2305 . . . . . 6 (x A (x = ∅ y A x = suc y) ↔ x(x A → (x = ∅ y A x = suc y)))
53, 4sylibr 137 . . . . 5 (x(x A ↔ (x = ∅ y A x = suc y)) → x A (x = ∅ y A x = suc y))
6 bj-inf2vn.1 . . . . . 6 BOUNDED A
7 bdcv 9303 . . . . . 6 BOUNDED z
86, 7bj-inf2vnlem3 9426 . . . . 5 (x A (x = ∅ y A x = suc y) → (Ind zAz))
95, 8syl 14 . . . 4 (x(x A ↔ (x = ∅ y A x = suc y)) → (Ind zAz))
109alrimiv 1751 . . 3 (x(x A ↔ (x = ∅ y A x = suc y)) → z(Ind zAz))
111, 10jca 290 . 2 (x(x A ↔ (x = ∅ y A x = suc y)) → (Ind A z(Ind zAz)))
12 bj-om 9394 . 2 (A 𝑉 → (A = 𝜔 ↔ (Ind A z(Ind zAz))))
1311, 12syl5ibr 145 1 (A 𝑉 → (x(x A ↔ (x = ∅ y A x = suc y)) → A = 𝜔))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  ∀wal 1240   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911  ∅c0 3218  suc csuc 4068  𝜔com 4256  BOUNDED wbdc 9295  Ind wind 9383 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-in1 544  ax-in2 545  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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9268  ax-bdim 9269  ax-bdor 9271  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339  ax-bdsetind 9422 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-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9296  df-bj-ind 9384 This theorem is referenced by:  bj-omex2  9431  bj-nn0sucALT  9432
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