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Theorem bj-nntrans 9411
Description: A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nntrans (A 𝜔 → (B ABA))

Proof of Theorem bj-nntrans
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3316 . . 3 x x ⊆ ∅
2 df-suc 4074 . . . . . . 7 suc z = (z ∪ {z})
32eleq2i 2101 . . . . . 6 (x suc zx (z ∪ {z}))
4 elun 3078 . . . . . . 7 (x (z ∪ {z}) ↔ (x z x {z}))
5 sssucid 4118 . . . . . . . . . 10 z ⊆ suc z
6 sstr2 2946 . . . . . . . . . 10 (xz → (z ⊆ suc zx ⊆ suc z))
75, 6mpi 15 . . . . . . . . 9 (xzx ⊆ suc z)
87imim2i 12 . . . . . . . 8 ((x zxz) → (x zx ⊆ suc z))
9 elsni 3391 . . . . . . . . . 10 (x {z} → x = z)
109, 5syl6eqss 2989 . . . . . . . . 9 (x {z} → x ⊆ suc z)
1110a1i 9 . . . . . . . 8 ((x zxz) → (x {z} → x ⊆ suc z))
128, 11jaod 636 . . . . . . 7 ((x zxz) → ((x z x {z}) → x ⊆ suc z))
134, 12syl5bi 141 . . . . . 6 ((x zxz) → (x (z ∪ {z}) → x ⊆ suc z))
143, 13syl5bi 141 . . . . 5 ((x zxz) → (x suc zx ⊆ suc z))
1514ralimi2 2375 . . . 4 (x z xzx suc zx ⊆ suc z)
1615rgenw 2370 . . 3 z 𝜔 (x z xzx suc zx ⊆ suc z)
17 bdcv 9303 . . . . . 6 BOUNDED y
1817bdss 9319 . . . . 5 BOUNDED xy
1918ax-bdal 9273 . . . 4 BOUNDED x y xy
20 nfv 1418 . . . 4 yx x ⊆ ∅
21 nfv 1418 . . . 4 yx z xz
22 nfv 1418 . . . 4 yx suc zx ⊆ suc z
23 sseq2 2961 . . . . . 6 (y = ∅ → (xyx ⊆ ∅))
2423raleqbi1dv 2507 . . . . 5 (y = ∅ → (x y xyx x ⊆ ∅))
2524biimprd 147 . . . 4 (y = ∅ → (x x ⊆ ∅ → x y xy))
26 sseq2 2961 . . . . . 6 (y = z → (xyxz))
2726raleqbi1dv 2507 . . . . 5 (y = z → (x y xyx z xz))
2827biimpd 132 . . . 4 (y = z → (x y xyx z xz))
29 sseq2 2961 . . . . . 6 (y = suc z → (xyx ⊆ suc z))
3029raleqbi1dv 2507 . . . . 5 (y = suc z → (x y xyx suc zx ⊆ suc z))
3130biimprd 147 . . . 4 (y = suc z → (x suc zx ⊆ suc zx y xy))
32 nfcv 2175 . . . 4 yA
33 nfv 1418 . . . 4 yx A xA
34 sseq2 2961 . . . . . 6 (y = A → (xyxA))
3534raleqbi1dv 2507 . . . . 5 (y = A → (x y xyx A xA))
3635biimpd 132 . . . 4 (y = A → (x y xyx A xA))
3719, 20, 21, 22, 25, 28, 31, 32, 33, 36bj-bdfindisg 9408 . . 3 ((x x ⊆ ∅ z 𝜔 (x z xzx suc zx ⊆ suc z)) → (A 𝜔 → x A xA))
381, 16, 37mp2an 402 . 2 (A 𝜔 → x A xA)
39 nfv 1418 . . 3 x BA
40 sseq1 2960 . . 3 (x = B → (xABA))
4139, 40rspc 2644 . 2 (B A → (x A xABA))
4238, 41syl5com 26 1 (A 𝜔 → (B ABA))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242   wcel 1390  wral 2300  cun 2909  wss 2911  c0 3218  {csn 3367  suc csuc 4068  𝜔com 4256
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-bndl 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-bdor 9271  ax-bdal 9273  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339  ax-infvn 9401
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 9386
This theorem is referenced by:  bj-nntrans2  9412  bj-nnelirr  9413  bj-nnen2lp  9414
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