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Theorem bj-nntrans 7173
 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 3301 . . 3 x x ⊆ ∅
2 df-suc 4057 . . . . . . 7 suc z = (z ∪ {z})
32eleq2i 2086 . . . . . 6 (x suc zx (z ∪ {z}))
4 elun 3061 . . . . . . 7 (x (z ∪ {z}) ↔ (x z x {z}))
5 sssucid 4101 . . . . . . . . . 10 z ⊆ suc z
6 sstr2 2929 . . . . . . . . . 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 3374 . . . . . . . . . 10 (x {z} → x = z)
109, 5syl6eqss 2972 . . . . . . . . 9 (x {z} → x ⊆ suc z)
1110a1i 9 . . . . . . . 8 ((x zxz) → (x {z} → x ⊆ suc z))
128, 11jaod 624 . . . . . . 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 2359 . . . 4 (x z xzx suc zx ⊆ suc z)
1615rgenw 2354 . . 3 z 𝜔 (x z xzx suc zx ⊆ suc z)
17 bdcv 7075 . . . . . 6 BOUNDED y
1817bdss 7091 . . . . 5 BOUNDED xy
1918ax-bdal 7045 . . . 4 BOUNDED x y xy
20 nfv 1402 . . . 4 yx x ⊆ ∅
21 nfv 1402 . . . 4 yx z xz
22 nfv 1402 . . . 4 yx suc zx ⊆ suc z
23 sseq2 2944 . . . . . 6 (y = ∅ → (xyx ⊆ ∅))
2423raleqbi1dv 2491 . . . . 5 (y = ∅ → (x y xyx x ⊆ ∅))
2524biimprd 147 . . . 4 (y = ∅ → (x x ⊆ ∅ → x y xy))
26 sseq2 2944 . . . . . 6 (y = z → (xyxz))
2726raleqbi1dv 2491 . . . . 5 (y = z → (x y xyx z xz))
2827biimpd 132 . . . 4 (y = z → (x y xyx z xz))
29 sseq2 2944 . . . . . 6 (y = suc z → (xyx ⊆ suc z))
3029raleqbi1dv 2491 . . . . 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 2160 . . . 4 yA
33 nfv 1402 . . . 4 yx A xA
34 sseq2 2944 . . . . . 6 (y = A → (xyxA))
3534raleqbi1dv 2491 . . . . 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 7170 . . 3 ((x x ⊆ ∅ z 𝜔 (x z xzx suc zx ⊆ suc z)) → (A 𝜔 → x A xA))
381, 16, 37mp2an 404 . 2 (A 𝜔 → x A xA)
39 nfv 1402 . . 3 x BA
40 sseq1 2943 . . 3 (x = B → (xABA))
4139, 40rspc 2627 . 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 616   = wceq 1228   ∈ wcel 1374  ∀wral 2284   ∪ cun 2892   ⊆ wss 2894  ∅c0 3201  {csn 3350  suc csuc 4051  𝜔com 4240 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdal 7045  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111  ax-infvn 7163 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150 This theorem is referenced by:  bj-nntrans2  7174  bj-nnelirr  7175  bj-nnen2lp  7176
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