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Theorem bj-nntrans 10435
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 3349 . . 3 𝑥 ∈ ∅ 𝑥 ⊆ ∅
2 df-suc 4135 . . . . . . 7 suc 𝑧 = (𝑧 ∪ {𝑧})
32eleq2i 2120 . . . . . 6 (𝑥 ∈ suc 𝑧𝑥 ∈ (𝑧 ∪ {𝑧}))
4 elun 3111 . . . . . . 7 (𝑥 ∈ (𝑧 ∪ {𝑧}) ↔ (𝑥𝑧𝑥 ∈ {𝑧}))
5 sssucid 4179 . . . . . . . . . 10 𝑧 ⊆ suc 𝑧
6 sstr2 2979 . . . . . . . . . 10 (𝑥𝑧 → (𝑧 ⊆ suc 𝑧𝑥 ⊆ suc 𝑧))
75, 6mpi 15 . . . . . . . . 9 (𝑥𝑧𝑥 ⊆ suc 𝑧)
87imim2i 12 . . . . . . . 8 ((𝑥𝑧𝑥𝑧) → (𝑥𝑧𝑥 ⊆ suc 𝑧))
9 elsni 3420 . . . . . . . . . 10 (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
109, 5syl6eqss 3022 . . . . . . . . 9 (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧)
1110a1i 9 . . . . . . . 8 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧))
128, 11jaod 647 . . . . . . 7 ((𝑥𝑧𝑥𝑧) → ((𝑥𝑧𝑥 ∈ {𝑧}) → 𝑥 ⊆ suc 𝑧))
134, 12syl5bi 145 . . . . . 6 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ (𝑧 ∪ {𝑧}) → 𝑥 ⊆ suc 𝑧))
143, 13syl5bi 145 . . . . 5 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
1514ralimi2 2398 . . . 4 (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
1615rgenw 2393 . . 3 𝑧 ∈ ω (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
17 bdcv 10327 . . . . . 6 BOUNDED 𝑦
1817bdss 10343 . . . . 5 BOUNDED 𝑥𝑦
1918ax-bdal 10297 . . . 4 BOUNDED𝑥𝑦 𝑥𝑦
20 nfv 1437 . . . 4 𝑦𝑥 ∈ ∅ 𝑥 ⊆ ∅
21 nfv 1437 . . . 4 𝑦𝑥𝑧 𝑥𝑧
22 nfv 1437 . . . 4 𝑦𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧
23 sseq2 2994 . . . . . 6 (𝑦 = ∅ → (𝑥𝑦𝑥 ⊆ ∅))
2423raleqbi1dv 2530 . . . . 5 (𝑦 = ∅ → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥 ∈ ∅ 𝑥 ⊆ ∅))
2524biimprd 151 . . . 4 (𝑦 = ∅ → (∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ → ∀𝑥𝑦 𝑥𝑦))
26 sseq2 2994 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
2726raleqbi1dv 2530 . . . . 5 (𝑦 = 𝑧 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥𝑧 𝑥𝑧))
2827biimpd 136 . . . 4 (𝑦 = 𝑧 → (∀𝑥𝑦 𝑥𝑦 → ∀𝑥𝑧 𝑥𝑧))
29 sseq2 2994 . . . . . 6 (𝑦 = suc 𝑧 → (𝑥𝑦𝑥 ⊆ suc 𝑧))
3029raleqbi1dv 2530 . . . . 5 (𝑦 = suc 𝑧 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
3130biimprd 151 . . . 4 (𝑦 = suc 𝑧 → (∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧 → ∀𝑥𝑦 𝑥𝑦))
32 nfcv 2194 . . . 4 𝑦𝐴
33 nfv 1437 . . . 4 𝑦𝑥𝐴 𝑥𝐴
34 sseq2 2994 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
3534raleqbi1dv 2530 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥𝐴 𝑥𝐴))
3635biimpd 136 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝑥𝑦 → ∀𝑥𝐴 𝑥𝐴))
3719, 20, 21, 22, 25, 28, 31, 32, 33, 36bj-bdfindisg 10432 . . 3 ((∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ ∧ ∀𝑧 ∈ ω (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)) → (𝐴 ∈ ω → ∀𝑥𝐴 𝑥𝐴))
381, 16, 37mp2an 410 . 2 (𝐴 ∈ ω → ∀𝑥𝐴 𝑥𝐴)
39 nfv 1437 . . 3 𝑥 𝐵𝐴
40 sseq1 2993 . . 3 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
4139, 40rspc 2667 . 2 (𝐵𝐴 → (∀𝑥𝐴 𝑥𝐴𝐵𝐴))
4238, 41syl5com 29 1 (𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639   = wceq 1259  wcel 1409  wral 2323  cun 2942  wss 2944  c0 3251  {csn 3402  suc csuc 4129  ωcom 4340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910  ax-pr 3971  ax-un 4197  ax-bd0 10292  ax-bdor 10295  ax-bdal 10297  ax-bdex 10298  ax-bdeq 10299  ax-bdel 10300  ax-bdsb 10301  ax-bdsep 10363  ax-infvn 10425
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-pr 3409  df-uni 3608  df-int 3643  df-suc 4135  df-iom 4341  df-bdc 10320  df-bj-ind 10410
This theorem is referenced by:  bj-nntrans2  10436  bj-nnelirr  10437  bj-nnen2lp  10438
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