Theorem onm 4521

Description: The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)

Assertion

Ref	Expression
onm	⊢ ∃𝑥 𝑥 ∈ On

Proof of Theorem onm

Step	Hyp	Ref	Expression
1		0elon 4512	. . 3 ⊢ ∅ ∈ On
2		0ex 4236	. . . 4 ⊢ ∅ ∈ V
3		eleq1 2295	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 ∈ On ↔ ∅ ∈ On))
4	2, 3	ceqsexv 2852	. . 3 ⊢ (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) ↔ ∅ ∈ On)
5	1, 4	mpbir 146	. 2 ⊢ ∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On)
6		exsimpr 1667	. 2 ⊢ (∃𝑥(𝑥 = ∅ ∧ 𝑥 ∈ On) → ∃𝑥 𝑥 ∈ On)
7	5, 6	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ∈ On

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∅c0 3507  Oncon0 4483

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-nul 4235

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-dif 3212  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-uni 3914  df-tr 4208  df-iord 4486  df-on 4488

This theorem is referenced by: (None)