Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-omtrans Structured version   GIF version

Theorem bj-omtrans 9390
Description: The set 𝜔 is transitive. A natural number is included in 𝜔.

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 9376 . . 3 𝜔 V
2 sseq2 2961 . . . . . 6 (𝑎 = 𝜔 → (y𝑎y ⊆ 𝜔))
3 sseq2 2961 . . . . . 6 (𝑎 = 𝜔 → (suc y𝑎 ↔ suc y ⊆ 𝜔))
42, 3imbi12d 223 . . . . 5 (𝑎 = 𝜔 → ((y𝑎 → suc y𝑎) ↔ (y ⊆ 𝜔 → suc y ⊆ 𝜔)))
54ralbidv 2320 . . . 4 (𝑎 = 𝜔 → (y 𝜔 (y𝑎 → suc y𝑎) ↔ y 𝜔 (y ⊆ 𝜔 → suc y ⊆ 𝜔)))
6 sseq2 2961 . . . . 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 3249 . . . 4 ∅ ⊆ 𝑎
10 bdcv 9283 . . . . . 6 BOUNDED 𝑎
1110bdss 9299 . . . . 5 BOUNDED x𝑎
12 nfv 1418 . . . . 5 x∅ ⊆ 𝑎
13 nfv 1418 . . . . 5 x y𝑎
14 nfv 1418 . . . . 5 x suc y𝑎
15 sseq1 2960 . . . . . 6 (x = ∅ → (x𝑎 ↔ ∅ ⊆ 𝑎))
1615biimprd 147 . . . . 5 (x = ∅ → (∅ ⊆ 𝑎x𝑎))
17 sseq1 2960 . . . . . 6 (x = y → (x𝑎y𝑎))
1817biimpd 132 . . . . 5 (x = y → (x𝑎y𝑎))
19 sseq1 2960 . . . . . 6 (x = suc y → (x𝑎 ↔ suc y𝑎))
2019biimprd 147 . . . . 5 (x = suc y → (suc y𝑎x𝑎))
21 nfcv 2175 . . . . 5 xA
22 nfv 1418 . . . . 5 x A𝑎
23 sseq1 2960 . . . . . 6 (x = A → (x𝑎A𝑎))
2423biimpd 132 . . . . 5 (x = A → (x𝑎A𝑎))
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 9382 . . . 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 2602 . 2 (y 𝜔 (y ⊆ 𝜔 → suc y ⊆ 𝜔) → (A 𝜔 → A ⊆ 𝜔))
28 df-suc 4074 . . . 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 3500 . . . . 5 ((y 𝜔 y ⊆ 𝜔) → {y} ⊆ 𝜔)
3229, 31unssd 3113 . . . 4 ((y 𝜔 y ⊆ 𝜔) → (y ∪ {y}) ⊆ 𝜔)
3328, 32syl5eqss 2983 . . 3 ((y 𝜔 y ⊆ 𝜔) → suc y ⊆ 𝜔)
3433ex 108 . 2 (y 𝜔 → (y ⊆ 𝜔 → suc y ⊆ 𝜔))
3527, 34mprg 2372 1 (A 𝜔 → A ⊆ 𝜔)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wral 2300  cun 2909  wss 2911  c0 3218  {csn 3367  suc csuc 4068  𝜔com 4256
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9248  ax-bdor 9251  ax-bdal 9253  ax-bdex 9254  ax-bdeq 9255  ax-bdel 9256  ax-bdsb 9257  ax-bdsep 9319  ax-infvn 9375
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9276  df-bj-ind 9362
This theorem is referenced by:  bj-omtrans2  9391  bj-nn0suc  9394
  Copyright terms: Public domain W3C validator