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Theorem bj-omtrans 10440
Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4355.

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 10426 . . 3 ω ∈ V
2 sseq2 2994 . . . . . 6 (𝑎 = ω → (𝑦𝑎𝑦 ⊆ ω))
3 sseq2 2994 . . . . . 6 (𝑎 = ω → (suc 𝑦𝑎 ↔ suc 𝑦 ⊆ ω))
42, 3imbi12d 227 . . . . 5 (𝑎 = ω → ((𝑦𝑎 → suc 𝑦𝑎) ↔ (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
54ralbidv 2343 . . . 4 (𝑎 = ω → (∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6 sseq2 2994 . . . . 5 (𝑎 = ω → (𝐴𝑎𝐴 ⊆ ω))
76imbi2d 223 . . . 4 (𝑎 = ω → ((𝐴 ∈ ω → 𝐴𝑎) ↔ (𝐴 ∈ ω → 𝐴 ⊆ ω)))
85, 7imbi12d 227 . . 3 (𝑎 = ω → ((∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) → (𝐴 ∈ ω → 𝐴𝑎)) ↔ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))))
9 0ss 3282 . . . 4 ∅ ⊆ 𝑎
10 bdcv 10327 . . . . . 6 BOUNDED 𝑎
1110bdss 10343 . . . . 5 BOUNDED 𝑥𝑎
12 nfv 1437 . . . . 5 𝑥∅ ⊆ 𝑎
13 nfv 1437 . . . . 5 𝑥 𝑦𝑎
14 nfv 1437 . . . . 5 𝑥 suc 𝑦𝑎
15 sseq1 2993 . . . . . 6 (𝑥 = ∅ → (𝑥𝑎 ↔ ∅ ⊆ 𝑎))
1615biimprd 151 . . . . 5 (𝑥 = ∅ → (∅ ⊆ 𝑎𝑥𝑎))
17 sseq1 2993 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑎𝑦𝑎))
1817biimpd 136 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑎𝑦𝑎))
19 sseq1 2993 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑎 ↔ suc 𝑦𝑎))
2019biimprd 151 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝑦𝑎𝑥𝑎))
21 nfcv 2194 . . . . 5 𝑥𝐴
22 nfv 1437 . . . . 5 𝑥 𝐴𝑎
23 sseq1 2993 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑎𝐴𝑎))
2423biimpd 136 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑎𝐴𝑎))
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 10432 . . . 4 ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎)) → (𝐴 ∈ ω → 𝐴𝑎))
269, 25mpan 408 . . 3 (∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) → (𝐴 ∈ ω → 𝐴𝑎))
271, 8, 26vtocl 2625 . 2 (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28 df-suc 4135 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
29 simpr 107 . . . . 5 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ⊆ ω)
30 simpl 106 . . . . . 6 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ∈ ω)
3130snssd 3536 . . . . 5 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
3229, 31unssd 3146 . . . 4 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
3328, 32syl5eqss 3016 . . 3 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
3433ex 112 . 2 (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
3527, 34mprg 2395 1 (𝐴 ∈ ω → 𝐴 ⊆ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = 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-omtrans2  10441  bj-nnord  10442  bj-nn0suc  10448
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