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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 4129 | . . 3 ⊢ ∅ ∈ On | |
2 | 0ex 3884 | . . . 4 ⊢ ∅ ∈ V | |
3 | eleq1 2100 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 ∈ On ↔ ∅ ∈ On)) | |
4 | 2, 3 | ceqsexv 2593 | . . 3 ⊢ (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) ↔ ∅ ∈ On) |
5 | 1, 4 | mpbir 134 | . 2 ⊢ ∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) |
6 | exsimpr 1509 | . 2 ⊢ (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) → ∃𝑥 𝑥 ∈ On) | |
7 | 5, 6 | ax-mp 7 | 1 ⊢ ∃𝑥 𝑥 ∈ On |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∅c0 3224 Oncon0 4100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 |
This theorem is referenced by: (None) |
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