Theorem oneptr 43707

Description: The strict order on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
oneptr	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶))

Proof of Theorem oneptr

Step	Hyp	Ref	Expression
1		epweon 7725	. 2 ⊢ E We On
2		weso 5616	. 2 ⊢ ( E We On → E Or On)
3		sopo 5552	. . 3 ⊢ ( E Or On → E Po On)
4		potr 5546	. . . 4 ⊢ (( E Po On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶))
5	4	ex 413	. . 3 ⊢ ( E Po On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶)))
6	3, 5	syl 17	. 2 ⊢ ( E Or On → ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶)))
7	1, 2, 6	mp2b 10	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119   class class class wbr 5079   E cep 5524   Po wpo 5531   Or wor 5532   We wwe 5577  Oncon0 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5187  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6320  df-on 6321

This theorem is referenced by: (None)