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Theorem bj-nntrans 11013
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 → (𝐵𝐴𝐵𝐴))

Proof of Theorem bj-nntrans
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3359 . . 3 𝑥 ∈ ∅ 𝑥 ⊆ ∅
2 df-suc 4154 . . . . . . 7 suc 𝑧 = (𝑧 ∪ {𝑧})
32eleq2i 2149 . . . . . 6 (𝑥 ∈ suc 𝑧𝑥 ∈ (𝑧 ∪ {𝑧}))
4 elun 3123 . . . . . . 7 (𝑥 ∈ (𝑧 ∪ {𝑧}) ↔ (𝑥𝑧𝑥 ∈ {𝑧}))
5 sssucid 4198 . . . . . . . . . 10 𝑧 ⊆ suc 𝑧
6 sstr2 3015 . . . . . . . . . 10 (𝑥𝑧 → (𝑧 ⊆ suc 𝑧𝑥 ⊆ suc 𝑧))
75, 6mpi 15 . . . . . . . . 9 (𝑥𝑧𝑥 ⊆ suc 𝑧)
87imim2i 12 . . . . . . . 8 ((𝑥𝑧𝑥𝑧) → (𝑥𝑧𝑥 ⊆ suc 𝑧))
9 elsni 3434 . . . . . . . . . 10 (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
109, 5syl6eqss 3058 . . . . . . . . 9 (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧)
1110a1i 9 . . . . . . . 8 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧))
128, 11jaod 670 . . . . . . 7 ((𝑥𝑧𝑥𝑧) → ((𝑥𝑧𝑥 ∈ {𝑧}) → 𝑥 ⊆ suc 𝑧))
134, 12syl5bi 150 . . . . . 6 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ (𝑧 ∪ {𝑧}) → 𝑥 ⊆ suc 𝑧))
143, 13syl5bi 150 . . . . 5 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
1514ralimi2 2428 . . . 4 (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
1615rgenw 2423 . . 3 𝑧 ∈ ω (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
17 bdcv 10906 . . . . . 6 BOUNDED 𝑦
1817bdss 10922 . . . . 5 BOUNDED 𝑥𝑦
1918ax-bdal 10876 . . . 4 BOUNDED𝑥𝑦 𝑥𝑦
20 nfv 1462 . . . 4 𝑦𝑥 ∈ ∅ 𝑥 ⊆ ∅
21 nfv 1462 . . . 4 𝑦𝑥𝑧 𝑥𝑧
22 nfv 1462 . . . 4 𝑦𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧
23 sseq2 3030 . . . . . 6 (𝑦 = ∅ → (𝑥𝑦𝑥 ⊆ ∅))
2423raleqbi1dv 2562 . . . . 5 (𝑦 = ∅ → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥 ∈ ∅ 𝑥 ⊆ ∅))
2524biimprd 156 . . . 4 (𝑦 = ∅ → (∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ → ∀𝑥𝑦 𝑥𝑦))
26 sseq2 3030 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
2726raleqbi1dv 2562 . . . . 5 (𝑦 = 𝑧 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥𝑧 𝑥𝑧))
2827biimpd 142 . . . 4 (𝑦 = 𝑧 → (∀𝑥𝑦 𝑥𝑦 → ∀𝑥𝑧 𝑥𝑧))
29 sseq2 3030 . . . . . 6 (𝑦 = suc 𝑧 → (𝑥𝑦𝑥 ⊆ suc 𝑧))
3029raleqbi1dv 2562 . . . . 5 (𝑦 = suc 𝑧 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
3130biimprd 156 . . . 4 (𝑦 = suc 𝑧 → (∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧 → ∀𝑥𝑦 𝑥𝑦))
32 nfcv 2223 . . . 4 𝑦𝐴
33 nfv 1462 . . . 4 𝑦𝑥𝐴 𝑥𝐴
34 sseq2 3030 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
3534raleqbi1dv 2562 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥𝐴 𝑥𝐴))
3635biimpd 142 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝑥𝑦 → ∀𝑥𝐴 𝑥𝐴))
3719, 20, 21, 22, 25, 28, 31, 32, 33, 36bj-bdfindisg 11010 . . 3 ((∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ ∧ ∀𝑧 ∈ ω (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)) → (𝐴 ∈ ω → ∀𝑥𝐴 𝑥𝐴))
381, 16, 37mp2an 417 . 2 (𝐴 ∈ ω → ∀𝑥𝐴 𝑥𝐴)
39 nfv 1462 . . 3 𝑥 𝐵𝐴
40 sseq1 3029 . . 3 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
4139, 40rspc 2704 . 2 (𝐵𝐴 → (∀𝑥𝐴 𝑥𝐴𝐵𝐴))
4238, 41syl5com 29 1 (𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662   = wceq 1285  wcel 1434  wral 2353  cun 2980  wss 2982  c0 3267  {csn 3416  suc csuc 4148  ωcom 4359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3924  ax-pr 3992  ax-un 4216  ax-bd0 10871  ax-bdor 10874  ax-bdal 10876  ax-bdex 10877  ax-bdeq 10878  ax-bdel 10879  ax-bdsb 10880  ax-bdsep 10942  ax-infvn 11003
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-sn 3422  df-pr 3423  df-uni 3622  df-int 3657  df-suc 4154  df-iom 4360  df-bdc 10899  df-bj-ind 10989
This theorem is referenced by:  bj-nntrans2  11014  bj-nnelirr  11015  bj-nnen2lp  11016
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