Theorem onpsssuc 4295

Description: An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Assertion

Ref	Expression
onpsssuc	⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)

Proof of Theorem onpsssuc

Step	Hyp	Ref	Expression
1		elirr 4266	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
2		sucidg 4153	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
3		eleq2 2101	. . . . 5 ⊢ (𝐴 = suc 𝐴 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ suc 𝐴))
4	2, 3	syl5ibrcom 146	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 = suc 𝐴 → 𝐴 ∈ 𝐴))
5	1, 4	mtoi 590	. . 3 ⊢ (𝐴 ∈ On → ¬ 𝐴 = suc 𝐴)
6		sssucid 4152	. . 3 ⊢ 𝐴 ⊆ suc 𝐴
7	5, 6	jctil 295	. 2 ⊢ (𝐴 ∈ On → (𝐴 ⊆ suc 𝐴 ∧ ¬ 𝐴 = suc 𝐴))
8		dfpss2 3029	. 2 ⊢ (𝐴 ⊊ suc 𝐴 ↔ (𝐴 ⊆ suc 𝐴 ∧ ¬ 𝐴 = suc 𝐴))
9	7, 8	sylibr 137	1 ⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ⊆ wss 2917   ⊊ wpss 2918  Oncon0 4100  suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pss 2933  df-sn 3381  df-suc 4108

This theorem is referenced by: (None)