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Theorem bj-inf2vnlem1 9400
Description: Lemma for bj-inf2vn 9404. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inf2vnlem1 (x(x A ↔ (x = ∅ y A x = suc y)) → Ind A)
Distinct variable group:   x,A,y

Proof of Theorem bj-inf2vnlem1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 bi2 121 . . . . 5 ((x A ↔ (x = ∅ y A x = suc y)) → ((x = ∅ y A x = suc y) → x A))
2 ax-io 629 . . . . . 6 (((x = ∅ y A x = suc y) → x A) ↔ ((x = ∅ → x A) (y A x = suc yx A)))
32biimpi 113 . . . . 5 (((x = ∅ y A x = suc y) → x A) → ((x = ∅ → x A) (y A x = suc yx A)))
4 simpl 102 . . . . . 6 (((x = ∅ → x A) (y A x = suc yx A)) → (x = ∅ → x A))
5 eleq1 2097 . . . . . 6 (x = ∅ → (x A ↔ ∅ A))
64, 5mpbidi 140 . . . . 5 (((x = ∅ → x A) (y A x = suc yx A)) → (x = ∅ → ∅ A))
71, 3, 63syl 17 . . . 4 ((x A ↔ (x = ∅ y A x = suc y)) → (x = ∅ → ∅ A))
87alimi 1341 . . 3 (x(x A ↔ (x = ∅ y A x = suc y)) → x(x = ∅ → ∅ A))
9 exim 1487 . . 3 (x(x = ∅ → ∅ A) → (x x = ∅ → x A))
10 0ex 3875 . . . . . 6 V
1110isseti 2557 . . . . 5 x x = ∅
12 pm2.27 35 . . . . 5 (x x = ∅ → ((x x = ∅ → x A) → x A))
1311, 12ax-mp 7 . . . 4 ((x x = ∅ → x A) → x A)
14 bj-ex 9217 . . . 4 (x A → ∅ A)
1513, 14syl 14 . . 3 ((x x = ∅ → x A) → ∅ A)
168, 9, 153syl 17 . 2 (x(x A ↔ (x = ∅ y A x = suc y)) → ∅ A)
173simprd 107 . . . . . 6 (((x = ∅ y A x = suc y) → x A) → (y A x = suc yx A))
181, 17syl 14 . . . . 5 ((x A ↔ (x = ∅ y A x = suc y)) → (y A x = suc yx A))
1918alimi 1341 . . . 4 (x(x A ↔ (x = ∅ y A x = suc y)) → x(y A x = suc yx A))
20 eqid 2037 . . . . 5 suc z = suc z
21 suceq 4105 . . . . . . 7 (y = z → suc y = suc z)
2221eqeq2d 2048 . . . . . 6 (y = z → (suc z = suc y ↔ suc z = suc z))
2322rspcev 2650 . . . . 5 ((z A suc z = suc z) → y A suc z = suc y)
2420, 23mpan2 401 . . . 4 (z Ay A suc z = suc y)
25 vex 2554 . . . . . 6 z V
2625bj-sucex 9354 . . . . 5 suc z V
27 eqeq1 2043 . . . . . . 7 (x = suc z → (x = suc y ↔ suc z = suc y))
2827rexbidv 2321 . . . . . 6 (x = suc z → (y A x = suc yy A suc z = suc y))
29 eleq1 2097 . . . . . 6 (x = suc z → (x A ↔ suc z A))
3028, 29imbi12d 223 . . . . 5 (x = suc z → ((y A x = suc yx A) ↔ (y A suc z = suc y → suc z A)))
3126, 30spcv 2640 . . . 4 (x(y A x = suc yx A) → (y A suc z = suc y → suc z A))
3219, 24, 31syl2im 34 . . 3 (x(x A ↔ (x = ∅ y A x = suc y)) → (z A → suc z A))
3332ralrimiv 2385 . 2 (x(x A ↔ (x = ∅ y A x = suc y)) → z A suc z A)
34 df-bj-ind 9362 . 2 (Ind A ↔ (∅ A z A suc z A))
3516, 33, 34sylanbrc 394 1 (x(x A ↔ (x = ∅ y A x = suc y)) → Ind A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628  wal 1240   = wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301  c0 3218  suc csuc 4068  Ind wind 9361
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 9248  ax-bdor 9251  ax-bdex 9254  ax-bdeq 9255  ax-bdel 9256  ax-bdsb 9257  ax-bdsep 9319
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-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-suc 4074  df-bdc 9276  df-bj-ind 9362
This theorem is referenced by:  bj-inf2vn  9404  bj-inf2vn2  9405
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