Theorem bj-el2oss1o 16546

Description: Shorter proof of el2oss1o 6676 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-el2oss1o	⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)

Proof of Theorem bj-el2oss1o

Step	Hyp	Ref	Expression
1		1on 6654	. . . 4 ⊢ 1o ∈ On
2	1	ontrci 4548	. . 3 ⊢ Tr 1o
3		trsucss 4544	. . 3 ⊢ (Tr 1o → (𝐴 ∈ suc 1o → 𝐴 ⊆ 1o))
4	2, 3	ax-mp 5	. 2 ⊢ (𝐴 ∈ suc 1o → 𝐴 ⊆ 1o)
5		df-2o 6648	. 2 ⊢ 2o = suc 1o
6	4, 5	eleq2s 2327	1 ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)

Syntax hints:   → wi 4   ∈ wcel 2203   ⊆ wss 3211  Tr wtr 4208  suc csuc 4486  1_oc1o 6640  2_oc2o 6641

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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

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 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492  df-1o 6647  df-2o 6648

This theorem is referenced by: (None)