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Theorem bj-omtrans 14628
Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4605.

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 14614 . . 3 ω ∈ V
2 sseq2 3179 . . . . . 6 (𝑎 = ω → (𝑦𝑎𝑦 ⊆ ω))
3 sseq2 3179 . . . . . 6 (𝑎 = ω → (suc 𝑦𝑎 ↔ suc 𝑦 ⊆ ω))
42, 3imbi12d 234 . . . . 5 (𝑎 = ω → ((𝑦𝑎 → suc 𝑦𝑎) ↔ (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
54ralbidv 2477 . . . 4 (𝑎 = ω → (∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6 sseq2 3179 . . . . 5 (𝑎 = ω → (𝐴𝑎𝐴 ⊆ ω))
76imbi2d 230 . . . 4 (𝑎 = ω → ((𝐴 ∈ ω → 𝐴𝑎) ↔ (𝐴 ∈ ω → 𝐴 ⊆ ω)))
85, 7imbi12d 234 . . 3 (𝑎 = ω → ((∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) → (𝐴 ∈ ω → 𝐴𝑎)) ↔ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))))
9 0ss 3461 . . . 4 ∅ ⊆ 𝑎
10 bdcv 14520 . . . . . 6 BOUNDED 𝑎
1110bdss 14536 . . . . 5 BOUNDED 𝑥𝑎
12 nfv 1528 . . . . 5 𝑥∅ ⊆ 𝑎
13 nfv 1528 . . . . 5 𝑥 𝑦𝑎
14 nfv 1528 . . . . 5 𝑥 suc 𝑦𝑎
15 sseq1 3178 . . . . . 6 (𝑥 = ∅ → (𝑥𝑎 ↔ ∅ ⊆ 𝑎))
1615biimprd 158 . . . . 5 (𝑥 = ∅ → (∅ ⊆ 𝑎𝑥𝑎))
17 sseq1 3178 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑎𝑦𝑎))
1817biimpd 144 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑎𝑦𝑎))
19 sseq1 3178 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑎 ↔ suc 𝑦𝑎))
2019biimprd 158 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝑦𝑎𝑥𝑎))
21 nfcv 2319 . . . . 5 𝑥𝐴
22 nfv 1528 . . . . 5 𝑥 𝐴𝑎
23 sseq1 3178 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑎𝐴𝑎))
2423biimpd 144 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑎𝐴𝑎))
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 14620 . . . 4 ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎)) → (𝐴 ∈ ω → 𝐴𝑎))
269, 25mpan 424 . . 3 (∀𝑦 ∈ ω (𝑦𝑎 → suc 𝑦𝑎) → (𝐴 ∈ ω → 𝐴𝑎))
271, 8, 26vtocl 2791 . 2 (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28 df-suc 4371 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
29 simpr 110 . . . . 5 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ⊆ ω)
30 simpl 109 . . . . . 6 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ∈ ω)
3130snssd 3737 . . . . 5 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
3229, 31unssd 3311 . . . 4 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
3328, 32eqsstrid 3201 . . 3 ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
3433ex 115 . 2 (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
3527, 34mprg 2534 1 (𝐴 ∈ ω → 𝐴 ⊆ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  cun 3127  wss 3129  c0 3422  {csn 3592  suc csuc 4365  ωcom 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4129  ax-pr 4209  ax-un 4433  ax-bd0 14485  ax-bdor 14488  ax-bdal 14490  ax-bdex 14491  ax-bdeq 14492  ax-bdel 14493  ax-bdsb 14494  ax-bdsep 14556  ax-infvn 14613
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4371  df-iom 4590  df-bdc 14513  df-bj-ind 14599
This theorem is referenced by:  bj-omtrans2  14629  bj-nnord  14630  bj-nn0suc  14636
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