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Theorem bj-2inf 10392
 Description: Two formulations of the axiom of infinity (see ax-infvn 10396 and bj-omex 10397) . (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 2054 . . . 4 ω = ω
2 bj-om 10391 . . . 4 (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
31, 2mpbii 140 . . 3 (ω ∈ V → (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
4 bj-indeq 10383 . . . . 5 (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5 sseq1 2991 . . . . . . 7 (𝑥 = ω → (𝑥𝑦 ↔ ω ⊆ 𝑦))
65imbi2d 223 . . . . . 6 (𝑥 = ω → ((Ind 𝑦𝑥𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦)))
76albidv 1719 . . . . 5 (𝑥 = ω → (∀𝑦(Ind 𝑦𝑥𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
84, 7anbi12d 450 . . . 4 (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
98spcegv 2656 . . 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 2575 . . . . . 6 𝑥 ∈ V
12 bj-om 10391 . . . . . 6 (𝑥 ∈ V → (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))))
1311, 12ax-mp 7 . . . . 5 (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
1413biimpri 128 . . . 4 ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → 𝑥 = ω)
1514eximi 1505 . . 3 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ∃𝑥 𝑥 = ω)
16 isset 2576 . . 3 (ω ∈ V ↔ ∃𝑥 𝑥 = ω)
1715, 16sylibr 141 . 2 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ω ∈ V)
1810, 17impbii 121 1 (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  ∀wal 1255   = wceq 1257  ∃wex 1395   ∈ wcel 1407  Vcvv 2572   ⊆ wss 2942  ωcom 4338  Ind wind 10380 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-nul 3908  ax-pr 3969  ax-un 4195  ax-bd0 10263  ax-bdor 10266  ax-bdex 10269  ax-bdeq 10270  ax-bdel 10271  ax-bdsb 10272  ax-bdsep 10334 This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-rab 2330  df-v 2574  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-sn 3406  df-pr 3407  df-uni 3606  df-int 3641  df-suc 4133  df-iom 4339  df-bdc 10291  df-bj-ind 10381 This theorem is referenced by:  bj-omex  10397
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