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Theorem bj-omtrans 8340
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 8326 . . 3 𝜔 V
2 sseq2 2935 . . . . . 6 (𝑎 = 𝜔 → (y𝑎y ⊆ 𝜔))
3 sseq2 2935 . . . . . 6 (𝑎 = 𝜔 → (suc y𝑎 ↔ suc y ⊆ 𝜔))
42, 3imbi12d 223 . . . . 5 (𝑎 = 𝜔 → ((y𝑎 → suc y𝑎) ↔ (y ⊆ 𝜔 → suc y ⊆ 𝜔)))
54ralbidv 2295 . . . 4 (𝑎 = 𝜔 → (y 𝜔 (y𝑎 → suc y𝑎) ↔ y 𝜔 (y ⊆ 𝜔 → suc y ⊆ 𝜔)))
6 sseq2 2935 . . . . 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 3223 . . . 4 ∅ ⊆ 𝑎
10 bdcv 8233 . . . . . 6 BOUNDED 𝑎
1110bdss 8249 . . . . 5 BOUNDED x𝑎
12 nfv 1394 . . . . 5 x∅ ⊆ 𝑎
13 nfv 1394 . . . . 5 x y𝑎
14 nfv 1394 . . . . 5 x suc y𝑎
15 sseq1 2934 . . . . . 6 (x = ∅ → (x𝑎 ↔ ∅ ⊆ 𝑎))
1615biimprd 147 . . . . 5 (x = ∅ → (∅ ⊆ 𝑎x𝑎))
17 sseq1 2934 . . . . . 6 (x = y → (x𝑎y𝑎))
1817biimpd 132 . . . . 5 (x = y → (x𝑎y𝑎))
19 sseq1 2934 . . . . . 6 (x = suc y → (x𝑎 ↔ suc y𝑎))
2019biimprd 147 . . . . 5 (x = suc y → (suc y𝑎x𝑎))
21 nfcv 2151 . . . . 5 xA
22 nfv 1394 . . . . 5 x A𝑎
23 sseq1 2934 . . . . . 6 (x = A → (x𝑎A𝑎))
2423biimpd 132 . . . . 5 (x = A → (x𝑎A𝑎))
2511, 12, 13, 14, 16, 18, 20, 21, 22, 24bj-bdfindisg 8332 . . . 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 2576 . 2 (y 𝜔 (y ⊆ 𝜔 → suc y ⊆ 𝜔) → (A 𝜔 → A ⊆ 𝜔))
28 df-suc 4046 . . . 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 3472 . . . . 5 ((y 𝜔 y ⊆ 𝜔) → {y} ⊆ 𝜔)
3229, 31unssd 3087 . . . 4 ((y 𝜔 y ⊆ 𝜔) → (y ∪ {y}) ⊆ 𝜔)
3328, 32syl5eqss 2957 . . 3 ((y 𝜔 y ⊆ 𝜔) → suc y ⊆ 𝜔)
3433ex 108 . 2 (y 𝜔 → (y ⊆ 𝜔 → suc y ⊆ 𝜔))
3527, 34mprg 2347 1 (A 𝜔 → A ⊆ 𝜔)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223   wcel 1366  wral 2275  cun 2883  wss 2885  c0 3192  {csn 3339  suc csuc 4040  𝜔com 4228
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-nul 3846  ax-pr 3907  ax-un 4108  ax-bd0 8198  ax-bdor 8201  ax-bdal 8203  ax-bdex 8204  ax-bdeq 8205  ax-bdel 8206  ax-bdsb 8207  ax-bdsep 8269  ax-infvn 8325
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-rab 2284  df-v 2528  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-sn 3345  df-pr 3346  df-uni 3544  df-int 3579  df-suc 4046  df-iom 4229  df-bdc 8226  df-bj-ind 8312
This theorem is referenced by:  bj-omtrans2  8341  bj-nn0suc  8344
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