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Theorem bj-2inf 15148
Description: Two formulations of the axiom of infinity (see ax-infvn 15151 and bj-omex 15152) . (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 2189 . . . 4 ω = ω
2 bj-om 15147 . . . 4 (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
31, 2mpbii 148 . . 3 (ω ∈ V → (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
4 bj-indeq 15139 . . . . 5 (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5 sseq1 3193 . . . . . . 7 (𝑥 = ω → (𝑥𝑦 ↔ ω ⊆ 𝑦))
65imbi2d 230 . . . . . 6 (𝑥 = ω → ((Ind 𝑦𝑥𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦)))
76albidv 1835 . . . . 5 (𝑥 = ω → (∀𝑦(Ind 𝑦𝑥𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
84, 7anbi12d 473 . . . 4 (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
98spcegv 2840 . . 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 2755 . . . . . 6 𝑥 ∈ V
12 bj-om 15147 . . . . . 6 (𝑥 ∈ V → (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))))
1311, 12ax-mp 5 . . . . 5 (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
1413biimpri 133 . . . 4 ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → 𝑥 = ω)
1514eximi 1611 . . 3 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ∃𝑥 𝑥 = ω)
16 isset 2758 . . 3 (ω ∈ V ↔ ∃𝑥 𝑥 = ω)
1715, 16sylibr 134 . 2 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ω ∈ V)
1810, 17impbii 126 1 (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362   = wceq 1364  wex 1503  wcel 2160  Vcvv 2752  wss 3144  ωcom 4607  Ind wind 15136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4227  ax-un 4451  ax-bd0 15023  ax-bdor 15026  ax-bdex 15029  ax-bdeq 15030  ax-bdel 15031  ax-bdsb 15032  ax-bdsep 15094
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4389  df-iom 4608  df-bdc 15051  df-bj-ind 15137
This theorem is referenced by:  bj-omex  15152
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