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Theorem bj-omtrans 10081
Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4328.

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 10067 . . 3 ω ∈ V
2 sseq2 2967 . . . . . 6 (𝑎 = ω → (𝑦𝑎𝑦 ⊆ ω))
3 sseq2 2967 . . . . . 6 (𝑎 = ω → (suc 𝑦𝑎 ↔ suc 𝑦 ⊆ ω))
42, 3imbi12d 223 . . . . 5 (𝑎 = ω → ((𝑦𝑎 → suc 𝑦𝑎) ↔ (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
54ralbidv 2326 . . . 4 (𝑎 = ω → (∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6 sseq2 2967 . . . . 5 (𝑎 = ω → (𝐴𝑎𝐴 ⊆ ω))
76imbi2d 219 . . . 4 (𝑎 = ω → ((𝐴 ∈ ω → 𝐴𝑎) ↔ (𝐴 ∈ ω → 𝐴 ⊆ ω)))
85, 7imbi12d 223 . . 3 (𝑎 = ω → ((∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) → (𝐴 ∈ ω → 𝐴𝑎)) ↔ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))))
9 0ss 3255 . . . 4 ∅ ⊆ 𝑎
10 bdcv 9968 . . . . . 6 BOUNDED 𝑎
1110bdss 9984 . . . . 5 BOUNDED 𝑥𝑎
12 nfv 1421 . . . . 5 𝑥∅ ⊆ 𝑎
13 nfv 1421 . . . . 5 𝑥 𝑦𝑎
14 nfv 1421 . . . . 5 𝑥 suc 𝑦𝑎
15 sseq1 2966 . . . . . 6 (𝑥 = ∅ → (𝑥𝑎 ↔ ∅ ⊆ 𝑎))
1615biimprd 147 . . . . 5 (𝑥 = ∅ → (∅ ⊆ 𝑎𝑥𝑎))
17 sseq1 2966 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑎𝑦𝑎))
1817biimpd 132 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑎𝑦𝑎))
19 sseq1 2966 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑎 ↔ suc 𝑦𝑎))
2019biimprd 147 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝑦𝑎𝑥𝑎))
21 nfcv 2178 . . . . 5 𝑥𝐴
22 nfv 1421 . . . . 5 𝑥 𝐴𝑎
23 sseq1 2966 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑎𝐴𝑎))
2423biimpd 132 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑎𝐴𝑎))
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 10073 . . . 4 ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎)) → (𝐴 ∈ ω → 𝐴𝑎))
269, 25mpan 400 . . 3 (∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) → (𝐴 ∈ ω → 𝐴𝑎))
271, 8, 26vtocl 2608 . 2 (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28 df-suc 4108 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
29 simpr 103 . . . . 5 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ⊆ ω)
30 simpl 102 . . . . . 6 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ∈ ω)
3130snssd 3509 . . . . 5 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
3229, 31unssd 3119 . . . 4 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
3328, 32syl5eqss 2989 . . 3 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
3433ex 108 . 2 (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
3527, 34mprg 2378 1 (𝐴 ∈ ω → 𝐴 ⊆ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = 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 9933  ax-bdor 9936  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066
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 9961  df-bj-ind 10051
This theorem is referenced by:  bj-omtrans2  10082  bj-nnord  10083  bj-nn0suc  10089
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