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Theorem bj-2inf 9326
Description: Two formulations of the axiom of infinity (see ax-infvn 9329 and bj-omex 9330) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-2inf (𝜔 V ↔ x(Ind x y(Ind yxy)))
Distinct variable group:   x,y

Proof of Theorem bj-2inf
StepHypRef Expression
1 eqid 2037 . . . 4 𝜔 = 𝜔
2 bj-om 9325 . . . 4 (𝜔 V → (𝜔 = 𝜔 ↔ (Ind 𝜔 y(Ind y → 𝜔 ⊆ y))))
31, 2mpbii 136 . . 3 (𝜔 V → (Ind 𝜔 y(Ind y → 𝜔 ⊆ y)))
4 bj-indeq 9318 . . . . 5 (x = 𝜔 → (Ind x ↔ Ind 𝜔))
5 sseq1 2960 . . . . . . 7 (x = 𝜔 → (xy ↔ 𝜔 ⊆ y))
65imbi2d 219 . . . . . 6 (x = 𝜔 → ((Ind yxy) ↔ (Ind y → 𝜔 ⊆ y)))
76albidv 1702 . . . . 5 (x = 𝜔 → (y(Ind yxy) ↔ y(Ind y → 𝜔 ⊆ y)))
84, 7anbi12d 442 . . . 4 (x = 𝜔 → ((Ind x y(Ind yxy)) ↔ (Ind 𝜔 y(Ind y → 𝜔 ⊆ y))))
98spcegv 2635 . . 3 (𝜔 V → ((Ind 𝜔 y(Ind y → 𝜔 ⊆ y)) → x(Ind x y(Ind yxy))))
103, 9mpd 13 . 2 (𝜔 V → x(Ind x y(Ind yxy)))
11 vex 2554 . . . . . 6 x V
12 bj-om 9325 . . . . . 6 (x V → (x = 𝜔 ↔ (Ind x y(Ind yxy))))
1311, 12ax-mp 7 . . . . 5 (x = 𝜔 ↔ (Ind x y(Ind yxy)))
1413biimpri 124 . . . 4 ((Ind x y(Ind yxy)) → x = 𝜔)
1514eximi 1488 . . 3 (x(Ind x y(Ind yxy)) → x x = 𝜔)
16 isset 2555 . . 3 (𝜔 V ↔ x x = 𝜔)
1715, 16sylibr 137 . 2 (x(Ind x y(Ind yxy)) → 𝜔 V)
1810, 17impbii 117 1 (𝜔 V ↔ x(Ind x y(Ind yxy)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  wss 2911  𝜔com 4256  Ind wind 9315
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 9202  ax-bdor 9205  ax-bdex 9208  ax-bdeq 9209  ax-bdel 9210  ax-bdsb 9211  ax-bdsep 9273
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 9230  df-bj-ind 9316
This theorem is referenced by:  bj-omex  9330
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