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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 4392 | . . 3 ⊢ ∅ ∈ On | |
2 | 0ex 4130 | . . . 4 ⊢ ∅ ∈ V | |
3 | eleq1 2240 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 ∈ On ↔ ∅ ∈ On)) | |
4 | 2, 3 | ceqsexv 2776 | . . 3 ⊢ (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) ↔ ∅ ∈ On) |
5 | 1, 4 | mpbir 146 | . 2 ⊢ ∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) |
6 | exsimpr 1618 | . 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 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∅c0 3422 Oncon0 4363 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-nul 4129 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-uni 3810 df-tr 4102 df-iord 4366 df-on 4368 |
This theorem is referenced by: (None) |
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