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Theorem bj-inf2vnlem1 7384
Description: Lemma for bj-inf2vn 7388. 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 617 . . . . . 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 ax-ia1 99 . . . . . 6 (((x = ∅ → x A) (y A x = suc yx A)) → (x = ∅ → x A))
5 eleq1 2078 . . . . . 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 1320 . . 3 (x(x A ↔ (x = ∅ y A x = suc y)) → x(x = ∅ → ∅ A))
9 exim 1468 . . 3 (x(x = ∅ → ∅ A) → (x x = ∅ → x A))
10 0ex 3854 . . . . . 6 V
1110isseti 2537 . . . . 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 7201 . . . 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 1320 . . . 4 (x(x A ↔ (x = ∅ y A x = suc y)) → x(y A x = suc yx A))
20 eqid 2018 . . . . 5 suc z = suc z
21 suceq 4084 . . . . . . 7 (y = z → suc y = suc z)
2221eqeq2d 2029 . . . . . 6 (y = z → (suc z = suc y ↔ suc z = suc z))
2322rspcev 2629 . . . . 5 ((z A suc z = suc z) → y A suc z = suc y)
2420, 23mpan2 403 . . . 4 (z Ay A suc z = suc y)
25 vex 2534 . . . . . 6 z V
2625bj-sucex 7338 . . . . 5 suc z V
27 eqeq1 2024 . . . . . . 7 (x = suc z → (x = suc y ↔ suc z = suc y))
2827rexbidv 2301 . . . . . 6 (x = suc z → (y A x = suc yy A suc z = suc y))
29 eleq1 2078 . . . . . 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 2619 . . . 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 2365 . 2 (x(x A ↔ (x = ∅ y A x = suc y)) → z A suc z A)
34 df-bj-ind 7346 . 2 (Ind A ↔ (∅ A z A suc z A))
3516, 33, 34sylanbrc 396 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 616  wal 1224   = wceq 1226  wex 1358   wcel 1370  wral 2280  wrex 2281  c0 3197  suc csuc 4047  Ind wind 7345
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-nul 3853  ax-pr 3914  ax-un 4116  ax-bd0 7232  ax-bdor 7235  ax-bdex 7238  ax-bdeq 7239  ax-bdel 7240  ax-bdsb 7241  ax-bdsep 7303
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-un 2895  df-nul 3198  df-sn 3352  df-pr 3353  df-uni 3551  df-suc 4053  df-bdc 7260  df-bj-ind 7346
This theorem is referenced by:  bj-inf2vn  7388  bj-inf2vn2  7389
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