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Mirrors > Home > ILE Home > Th. List > onm | GIF version |
Description: The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.) |
Ref | Expression |
---|---|
onm | ⊢ ∃𝑥 𝑥 ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 4322 | . . 3 ⊢ ∅ ∈ On | |
2 | 0ex 4063 | . . . 4 ⊢ ∅ ∈ V | |
3 | eleq1 2203 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 ∈ On ↔ ∅ ∈ On)) | |
4 | 2, 3 | ceqsexv 2728 | . . 3 ⊢ (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) ↔ ∅ ∈ On) |
5 | 1, 4 | mpbir 145 | . 2 ⊢ ∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) |
6 | exsimpr 1598 | . 2 ⊢ (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) → ∃𝑥 𝑥 ∈ On) | |
7 | 5, 6 | ax-mp 5 | 1 ⊢ ∃𝑥 𝑥 ∈ On |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∅c0 3368 Oncon0 4293 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 |
This theorem is referenced by: (None) |
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