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Theorem bj-2inf 13166
Description: Two formulations of the axiom of infinity (see ax-infvn 13169 and bj-omex 13170) . (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 2139 . . . 4 ω = ω
2 bj-om 13165 . . . 4 (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
31, 2mpbii 147 . . 3 (ω ∈ V → (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
4 bj-indeq 13157 . . . . 5 (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5 sseq1 3120 . . . . . . 7 (𝑥 = ω → (𝑥𝑦 ↔ ω ⊆ 𝑦))
65imbi2d 229 . . . . . 6 (𝑥 = ω → ((Ind 𝑦𝑥𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦)))
76albidv 1796 . . . . 5 (𝑥 = ω → (∀𝑦(Ind 𝑦𝑥𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
84, 7anbi12d 464 . . . 4 (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
98spcegv 2774 . . 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 2689 . . . . . 6 𝑥 ∈ V
12 bj-om 13165 . . . . . 6 (𝑥 ∈ V → (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))))
1311, 12ax-mp 5 . . . . 5 (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
1413biimpri 132 . . . 4 ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → 𝑥 = ω)
1514eximi 1579 . . 3 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ∃𝑥 𝑥 = ω)
16 isset 2692 . . 3 (ω ∈ V ↔ ∃𝑥 𝑥 = ω)
1715, 16sylibr 133 . 2 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ω ∈ V)
1810, 17impbii 125 1 (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329   = wceq 1331  wex 1468  wcel 1480  Vcvv 2686  wss 3071  ωcom 4504  Ind wind 13154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13041  ax-bdor 13044  ax-bdex 13047  ax-bdeq 13048  ax-bdel 13049  ax-bdsb 13050  ax-bdsep 13112
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13069  df-bj-ind 13155
This theorem is referenced by:  bj-omex  13170
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