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Theorem bj-2inf 11174
Description: Two formulations of the axiom of infinity (see ax-infvn 11177 and bj-omex 11178) . (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 2083 . . . 4 ω = ω
2 bj-om 11173 . . . 4 (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
31, 2mpbii 146 . . 3 (ω ∈ V → (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
4 bj-indeq 11165 . . . . 5 (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5 sseq1 3031 . . . . . . 7 (𝑥 = ω → (𝑥𝑦 ↔ ω ⊆ 𝑦))
65imbi2d 228 . . . . . 6 (𝑥 = ω → ((Ind 𝑦𝑥𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦)))
76albidv 1747 . . . . 5 (𝑥 = ω → (∀𝑦(Ind 𝑦𝑥𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
84, 7anbi12d 457 . . . 4 (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
98spcegv 2697 . . 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 2615 . . . . . 6 𝑥 ∈ V
12 bj-om 11173 . . . . . 6 (𝑥 ∈ V → (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))))
1311, 12ax-mp 7 . . . . 5 (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
1413biimpri 131 . . . 4 ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → 𝑥 = ω)
1514eximi 1532 . . 3 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ∃𝑥 𝑥 = ω)
16 isset 2616 . . 3 (ω ∈ V ↔ ∃𝑥 𝑥 = ω)
1715, 16sylibr 132 . 2 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ω ∈ V)
1810, 17impbii 124 1 (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  Vcvv 2612  wss 2984  ωcom 4367  Ind wind 11162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3930  ax-pr 3999  ax-un 4223  ax-bd0 11045  ax-bdor 11048  ax-bdex 11051  ax-bdeq 11052  ax-bdel 11053  ax-bdsb 11054  ax-bdsep 11116
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-suc 4161  df-iom 4368  df-bdc 11073  df-bj-ind 11163
This theorem is referenced by:  bj-omex  11178
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