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Theorem bj-2inf 13935
Description: Two formulations of the axiom of infinity (see ax-infvn 13938 and bj-omex 13939) . (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 2170 . . . 4 ω = ω
2 bj-om 13934 . . . 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 13926 . . . . 5 (𝑥 = ω → (Ind 𝑥 ↔ Ind ω))
5 sseq1 3170 . . . . . . 7 (𝑥 = ω → (𝑥𝑦 ↔ ω ⊆ 𝑦))
65imbi2d 229 . . . . . 6 (𝑥 = ω → ((Ind 𝑦𝑥𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦)))
76albidv 1817 . . . . 5 (𝑥 = ω → (∀𝑦(Ind 𝑦𝑥𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))
84, 7anbi12d 470 . . . 4 (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))))
98spcegv 2818 . . 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 2733 . . . . . 6 𝑥 ∈ V
12 bj-om 13934 . . . . . 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 1593 . . 3 (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)) → ∃𝑥 𝑥 = ω)
16 isset 2736 . . 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 1346   = wceq 1348  wex 1485  wcel 2141  Vcvv 2730  wss 3121  ωcom 4572  Ind wind 13923
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4113  ax-pr 4192  ax-un 4416  ax-bd0 13810  ax-bdor 13813  ax-bdex 13816  ax-bdeq 13817  ax-bdel 13818  ax-bdsb 13819  ax-bdsep 13881
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3587  df-pr 3588  df-uni 3795  df-int 3830  df-suc 4354  df-iom 4573  df-bdc 13838  df-bj-ind 13924
This theorem is referenced by:  bj-omex  13939
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