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Theorem bj-omtrans 11036
Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4374.

The idea is to use bounded induction with the formula 𝑥 ⊆ ω. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with 𝑥𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)

Assertion
Ref Expression
bj-omtrans (𝐴 ∈ ω → 𝐴 ⊆ ω)

Proof of Theorem bj-omtrans
Dummy variables 𝑥 𝑎 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-omex 11022 . . 3 ω ∈ V
2 sseq2 3030 . . . . . 6 (𝑎 = ω → (𝑦𝑎𝑦 ⊆ ω))
3 sseq2 3030 . . . . . 6 (𝑎 = ω → (suc 𝑦𝑎 ↔ suc 𝑦 ⊆ ω))
42, 3imbi12d 232 . . . . 5 (𝑎 = ω → ((𝑦𝑎 → suc 𝑦𝑎) ↔ (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
54ralbidv 2373 . . . 4 (𝑎 = ω → (∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6 sseq2 3030 . . . . 5 (𝑎 = ω → (𝐴𝑎𝐴 ⊆ ω))
76imbi2d 228 . . . 4 (𝑎 = ω → ((𝐴 ∈ ω → 𝐴𝑎) ↔ (𝐴 ∈ ω → 𝐴 ⊆ ω)))
85, 7imbi12d 232 . . 3 (𝑎 = ω → ((∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) → (𝐴 ∈ ω → 𝐴𝑎)) ↔ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))))
9 0ss 3298 . . . 4 ∅ ⊆ 𝑎
10 bdcv 10924 . . . . . 6 BOUNDED 𝑎
1110bdss 10940 . . . . 5 BOUNDED 𝑥𝑎
12 nfv 1462 . . . . 5 𝑥∅ ⊆ 𝑎
13 nfv 1462 . . . . 5 𝑥 𝑦𝑎
14 nfv 1462 . . . . 5 𝑥 suc 𝑦𝑎
15 sseq1 3029 . . . . . 6 (𝑥 = ∅ → (𝑥𝑎 ↔ ∅ ⊆ 𝑎))
1615biimprd 156 . . . . 5 (𝑥 = ∅ → (∅ ⊆ 𝑎𝑥𝑎))
17 sseq1 3029 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑎𝑦𝑎))
1817biimpd 142 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑎𝑦𝑎))
19 sseq1 3029 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑎 ↔ suc 𝑦𝑎))
2019biimprd 156 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝑦𝑎𝑥𝑎))
21 nfcv 2223 . . . . 5 𝑥𝐴
22 nfv 1462 . . . . 5 𝑥 𝐴𝑎
23 sseq1 3029 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑎𝐴𝑎))
2423biimpd 142 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑎𝐴𝑎))
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 11028 . . . 4 ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎)) → (𝐴 ∈ ω → 𝐴𝑎))
269, 25mpan 415 . . 3 (∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) → (𝐴 ∈ ω → 𝐴𝑎))
271, 8, 26vtocl 2662 . 2 (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28 df-suc 4154 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
29 simpr 108 . . . . 5 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ⊆ ω)
30 simpl 107 . . . . . 6 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ∈ ω)
3130snssd 3550 . . . . 5 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
3229, 31unssd 3158 . . . 4 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
3328, 32syl5eqss 3052 . . 3 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
3433ex 113 . 2 (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
3527, 34mprg 2425 1 (𝐴 ∈ ω → 𝐴 ⊆ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = 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 10889  ax-bdor 10892  ax-bdal 10894  ax-bdex 10895  ax-bdeq 10896  ax-bdel 10897  ax-bdsb 10898  ax-bdsep 10960  ax-infvn 11021
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 10917  df-bj-ind 11007
This theorem is referenced by:  bj-omtrans2  11037  bj-nnord  11038  bj-nn0suc  11044
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