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Theorem bj-2inf 7307
 Description: Two formulations of the axiom of infinity (see ax-infvn 7310 and bj-omex 7311) . (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 2022 . . . 4 𝜔 = 𝜔
2 bj-om 7306 . . . 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 7299 . . . . 5 (x = 𝜔 → (Ind x ↔ Ind 𝜔))
5 sseq1 2943 . . . . . . 7 (x = 𝜔 → (xy ↔ 𝜔 ⊆ y))
65imbi2d 219 . . . . . 6 (x = 𝜔 → ((Ind yxy) ↔ (Ind y → 𝜔 ⊆ y)))
76albidv 1687 . . . . 5 (x = 𝜔 → (y(Ind yxy) ↔ y(Ind y → 𝜔 ⊆ y)))
84, 7anbi12d 445 . . . 4 (x = 𝜔 → ((Ind x y(Ind yxy)) ↔ (Ind 𝜔 y(Ind y → 𝜔 ⊆ y))))
98spcegv 2618 . . 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 2538 . . . . . 6 x V
12 bj-om 7306 . . . . . 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 1473 . . 3 (x(Ind x y(Ind yxy)) → x x = 𝜔)
16 isset 2539 . . 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 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535   ⊆ wss 2894  𝜔com 4240  Ind wind 7296 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7187  ax-bdor 7190  ax-bdex 7193  ax-bdeq 7194  ax-bdel 7195  ax-bdsb 7196  ax-bdsep 7258 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7215  df-bj-ind 7297 This theorem is referenced by:  bj-omex  7311
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