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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-2inf | GIF version |
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.) |
Ref | Expression |
---|---|
bj-2inf | ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2170 | . . . 4 ⊢ ω = ω | |
2 | bj-om 13934 | . . . 4 ⊢ (ω ∈ V → (ω = ω ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))) | |
3 | 1, 2 | mpbii 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 → (𝑥 ⊆ 𝑦 ↔ ω ⊆ 𝑦)) | |
6 | 5 | imbi2d 229 | . . . . . 6 ⊢ (𝑥 = ω → ((Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ (Ind 𝑦 → ω ⊆ 𝑦))) |
7 | 6 | albidv 1817 | . . . . 5 ⊢ (𝑥 = ω → (∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦) ↔ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦))) |
8 | 4, 7 | anbi12d 470 | . . . 4 ⊢ (𝑥 = ω → ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) ↔ (Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)))) |
9 | 8 | spcegv 2818 | . . 3 ⊢ (ω ∈ V → ((Ind ω ∧ ∀𝑦(Ind 𝑦 → ω ⊆ 𝑦)) → ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))) |
10 | 3, 9 | mpd 13 | . 2 ⊢ (ω ∈ V → ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) |
11 | vex 2733 | . . . . . 6 ⊢ 𝑥 ∈ V | |
12 | bj-om 13934 | . . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (𝑥 = ω ↔ (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) |
14 | 13 | biimpri 132 | . . . 4 ⊢ ((Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → 𝑥 = ω) |
15 | 14 | eximi 1593 | . . 3 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ∃𝑥 𝑥 = ω) |
16 | isset 2736 | . . 3 ⊢ (ω ∈ V ↔ ∃𝑥 𝑥 = ω) | |
17 | 15, 16 | sylibr 133 | . 2 ⊢ (∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) → ω ∈ V) |
18 | 10, 17 | impbii 125 | 1 ⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) |
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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