Theorem onelssex 6374

Description: Ordinal less than is equivalent to having an ordinal between them. (Contributed by Scott Fenton, 8-Aug-2024.)

Assertion

Ref	Expression
onelssex	⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏

Proof of Theorem onelssex

Step	Hyp	Ref	Expression
1		ssid 3958	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		sseq2 3962	. . . 4 ⊢ (𝑏 = 𝐴 → (𝐴 ⊆ 𝑏 ↔ 𝐴 ⊆ 𝐴))
3	2	rspcev 3578	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐴) → ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏)
4	1, 3	mpan2 692	. 2 ⊢ (𝐴 ∈ 𝐶 → ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏)
5		ontr2 6373	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝑏 ∧ 𝑏 ∈ 𝐶) → 𝐴 ∈ 𝐶))
6	5	expcomd 416	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝑏 ∈ 𝐶 → (𝐴 ⊆ 𝑏 → 𝐴 ∈ 𝐶)))
7	6	rexlimdv 3137	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏 → 𝐴 ∈ 𝐶))
8	4, 7	impbid2 226	1 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903  Oncon0 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329

This theorem is referenced by:  madebdayim  27896  madebdaylemold  27906