Theorem bj-el2oss1o 15910

Description: Shorter proof of el2oss1o 6552 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 6532	. . . 4 ⊢ 1o ∈ On
2	1	ontrci 4492	. . 3 ⊢ Tr 1o
3		trsucss 4488	. . 3 ⊢ (Tr 1o → (𝐴 ∈ suc 1o → 𝐴 ⊆ 1o))
4	2, 3	ax-mp 5	. 2 ⊢ (𝐴 ∈ suc 1o → 𝐴 ⊆ 1o)
5		df-2o 6526	. 2 ⊢ 2o = suc 1o
6	4, 5	eleq2s 2302	1 ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)

Syntax hints:   → wi 4   ∈ wcel 2178   ⊆ wss 3174  Tr wtr 4158  suc csuc 4430  1_oc1o 6518  2_oc2o 6519

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436  df-1o 6525  df-2o 6526

This theorem is referenced by: (None)