Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-nntrans GIF version

Theorem bj-nntrans 12960
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 3432 . . 3 𝑥 ∈ ∅ 𝑥 ⊆ ∅
2 df-suc 4261 . . . . . . 7 suc 𝑧 = (𝑧 ∪ {𝑧})
32eleq2i 2182 . . . . . 6 (𝑥 ∈ suc 𝑧𝑥 ∈ (𝑧 ∪ {𝑧}))
4 elun 3185 . . . . . . 7 (𝑥 ∈ (𝑧 ∪ {𝑧}) ↔ (𝑥𝑧𝑥 ∈ {𝑧}))
5 sssucid 4305 . . . . . . . . . 10 𝑧 ⊆ suc 𝑧
6 sstr2 3072 . . . . . . . . . 10 (𝑥𝑧 → (𝑧 ⊆ suc 𝑧𝑥 ⊆ suc 𝑧))
75, 6mpi 15 . . . . . . . . 9 (𝑥𝑧𝑥 ⊆ suc 𝑧)
87imim2i 12 . . . . . . . 8 ((𝑥𝑧𝑥𝑧) → (𝑥𝑧𝑥 ⊆ suc 𝑧))
9 elsni 3513 . . . . . . . . . 10 (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
109, 5eqsstrdi 3117 . . . . . . . . 9 (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧)
1110a1i 9 . . . . . . . 8 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧))
128, 11jaod 689 . . . . . . 7 ((𝑥𝑧𝑥𝑧) → ((𝑥𝑧𝑥 ∈ {𝑧}) → 𝑥 ⊆ suc 𝑧))
134, 12syl5bi 151 . . . . . 6 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ (𝑧 ∪ {𝑧}) → 𝑥 ⊆ suc 𝑧))
143, 13syl5bi 151 . . . . 5 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
1514ralimi2 2467 . . . 4 (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
1615rgenw 2462 . . 3 𝑧 ∈ ω (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
17 bdcv 12857 . . . . . 6 BOUNDED 𝑦
1817bdss 12873 . . . . 5 BOUNDED 𝑥𝑦
1918ax-bdal 12827 . . . 4 BOUNDED𝑥𝑦 𝑥𝑦
20 nfv 1491 . . . 4 𝑦𝑥 ∈ ∅ 𝑥 ⊆ ∅
21 nfv 1491 . . . 4 𝑦𝑥𝑧 𝑥𝑧
22 nfv 1491 . . . 4 𝑦𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧
23 sseq2 3089 . . . . . 6 (𝑦 = ∅ → (𝑥𝑦𝑥 ⊆ ∅))
2423raleqbi1dv 2609 . . . . 5 (𝑦 = ∅ → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥 ∈ ∅ 𝑥 ⊆ ∅))
2524biimprd 157 . . . 4 (𝑦 = ∅ → (∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ → ∀𝑥𝑦 𝑥𝑦))
26 sseq2 3089 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
2726raleqbi1dv 2609 . . . . 5 (𝑦 = 𝑧 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥𝑧 𝑥𝑧))
2827biimpd 143 . . . 4 (𝑦 = 𝑧 → (∀𝑥𝑦 𝑥𝑦 → ∀𝑥𝑧 𝑥𝑧))
29 sseq2 3089 . . . . . 6 (𝑦 = suc 𝑧 → (𝑥𝑦𝑥 ⊆ suc 𝑧))
3029raleqbi1dv 2609 . . . . 5 (𝑦 = suc 𝑧 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
3130biimprd 157 . . . 4 (𝑦 = suc 𝑧 → (∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧 → ∀𝑥𝑦 𝑥𝑦))
32 nfcv 2256 . . . 4 𝑦𝐴
33 nfv 1491 . . . 4 𝑦𝑥𝐴 𝑥𝐴
34 sseq2 3089 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
3534raleqbi1dv 2609 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥𝐴 𝑥𝐴))
3635biimpd 143 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝑥𝑦 → ∀𝑥𝐴 𝑥𝐴))
3719, 20, 21, 22, 25, 28, 31, 32, 33, 36bj-bdfindisg 12957 . . 3 ((∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ ∧ ∀𝑧 ∈ ω (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)) → (𝐴 ∈ ω → ∀𝑥𝐴 𝑥𝐴))
381, 16, 37mp2an 420 . 2 (𝐴 ∈ ω → ∀𝑥𝐴 𝑥𝐴)
39 nfv 1491 . . 3 𝑥 𝐵𝐴
40 sseq1 3088 . . 3 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
4139, 40rspc 2755 . 2 (𝐵𝐴 → (∀𝑥𝐴 𝑥𝐴𝐵𝐴))
4238, 41syl5com 29 1 (𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 680   = wceq 1314  wcel 1463  wral 2391  cun 3037  wss 3039  c0 3331  {csn 3495  suc csuc 4255  ωcom 4472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12822  ax-bdor 12825  ax-bdal 12827  ax-bdex 12828  ax-bdeq 12829  ax-bdel 12830  ax-bdsb 12831  ax-bdsep 12893  ax-infvn 12950
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-bdc 12850  df-bj-ind 12936
This theorem is referenced by:  bj-nntrans2  12961  bj-nnelirr  12962  bj-nnen2lp  12963
  Copyright terms: Public domain W3C validator