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Theorem bj-nntrans 10050
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 3322 . . 3 𝑥 ∈ ∅ 𝑥 ⊆ ∅
2 df-suc 4108 . . . . . . 7 suc 𝑧 = (𝑧 ∪ {𝑧})
32eleq2i 2104 . . . . . 6 (𝑥 ∈ suc 𝑧𝑥 ∈ (𝑧 ∪ {𝑧}))
4 elun 3084 . . . . . . 7 (𝑥 ∈ (𝑧 ∪ {𝑧}) ↔ (𝑥𝑧𝑥 ∈ {𝑧}))
5 sssucid 4152 . . . . . . . . . 10 𝑧 ⊆ suc 𝑧
6 sstr2 2952 . . . . . . . . . 10 (𝑥𝑧 → (𝑧 ⊆ suc 𝑧𝑥 ⊆ suc 𝑧))
75, 6mpi 15 . . . . . . . . 9 (𝑥𝑧𝑥 ⊆ suc 𝑧)
87imim2i 12 . . . . . . . 8 ((𝑥𝑧𝑥𝑧) → (𝑥𝑧𝑥 ⊆ suc 𝑧))
9 elsni 3393 . . . . . . . . . 10 (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
109, 5syl6eqss 2995 . . . . . . . . 9 (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧)
1110a1i 9 . . . . . . . 8 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧))
128, 11jaod 637 . . . . . . 7 ((𝑥𝑧𝑥𝑧) → ((𝑥𝑧𝑥 ∈ {𝑧}) → 𝑥 ⊆ suc 𝑧))
134, 12syl5bi 141 . . . . . 6 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ (𝑧 ∪ {𝑧}) → 𝑥 ⊆ suc 𝑧))
143, 13syl5bi 141 . . . . 5 ((𝑥𝑧𝑥𝑧) → (𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
1514ralimi2 2381 . . . 4 (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
1615rgenw 2376 . . 3 𝑧 ∈ ω (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
17 bdcv 9942 . . . . . 6 BOUNDED 𝑦
1817bdss 9958 . . . . 5 BOUNDED 𝑥𝑦
1918ax-bdal 9912 . . . 4 BOUNDED𝑥𝑦 𝑥𝑦
20 nfv 1421 . . . 4 𝑦𝑥 ∈ ∅ 𝑥 ⊆ ∅
21 nfv 1421 . . . 4 𝑦𝑥𝑧 𝑥𝑧
22 nfv 1421 . . . 4 𝑦𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧
23 sseq2 2967 . . . . . 6 (𝑦 = ∅ → (𝑥𝑦𝑥 ⊆ ∅))
2423raleqbi1dv 2513 . . . . 5 (𝑦 = ∅ → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥 ∈ ∅ 𝑥 ⊆ ∅))
2524biimprd 147 . . . 4 (𝑦 = ∅ → (∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ → ∀𝑥𝑦 𝑥𝑦))
26 sseq2 2967 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
2726raleqbi1dv 2513 . . . . 5 (𝑦 = 𝑧 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥𝑧 𝑥𝑧))
2827biimpd 132 . . . 4 (𝑦 = 𝑧 → (∀𝑥𝑦 𝑥𝑦 → ∀𝑥𝑧 𝑥𝑧))
29 sseq2 2967 . . . . . 6 (𝑦 = suc 𝑧 → (𝑥𝑦𝑥 ⊆ suc 𝑧))
3029raleqbi1dv 2513 . . . . 5 (𝑦 = suc 𝑧 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
3130biimprd 147 . . . 4 (𝑦 = suc 𝑧 → (∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧 → ∀𝑥𝑦 𝑥𝑦))
32 nfcv 2178 . . . 4 𝑦𝐴
33 nfv 1421 . . . 4 𝑦𝑥𝐴 𝑥𝐴
34 sseq2 2967 . . . . . 6 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
3534raleqbi1dv 2513 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑦 𝑥𝑦 ↔ ∀𝑥𝐴 𝑥𝐴))
3635biimpd 132 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝑥𝑦 → ∀𝑥𝐴 𝑥𝐴))
3719, 20, 21, 22, 25, 28, 31, 32, 33, 36bj-bdfindisg 10047 . . 3 ((∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ ∧ ∀𝑧 ∈ ω (∀𝑥𝑧 𝑥𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)) → (𝐴 ∈ ω → ∀𝑥𝐴 𝑥𝐴))
381, 16, 37mp2an 402 . 2 (𝐴 ∈ ω → ∀𝑥𝐴 𝑥𝐴)
39 nfv 1421 . . 3 𝑥 𝐵𝐴
40 sseq1 2966 . . 3 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
4139, 40rspc 2650 . 2 (𝐵𝐴 → (∀𝑥𝐴 𝑥𝐴𝐵𝐴))
4238, 41syl5com 26 1 (𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 629   = wceq 1243  wcel 1393  wral 2306  cun 2915  wss 2917  c0 3224  {csn 3375  suc csuc 4102  ωcom 4313
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9907  ax-bdor 9910  ax-bdal 9912  ax-bdex 9913  ax-bdeq 9914  ax-bdel 9915  ax-bdsb 9916  ax-bdsep 9978  ax-infvn 10040
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9935  df-bj-ind 10025
This theorem is referenced by:  bj-nntrans2  10051  bj-nnelirr  10052  bj-nnen2lp  10053
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