Theorem orddif 4674

Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)

Assertion

Ref	Expression
orddif	⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))

Proof of Theorem orddif

Step	Hyp	Ref	Expression
1		orddisj 4673	. 2 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
2		disj3 3565	. . 3 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴}))
3		df-suc 4497	. . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	difeq1i 3337	. . . . 5 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴})
5		difun2 3593	. . . . 5 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴})
6	4, 5	eqtri 2255	. . . 4 ⊢ (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴})
7	6	eqeq2i 2245	. . 3 ⊢ (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴}))
8	2, 7	bitr4i 187	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴}))
9	1, 8	sylib 122	1 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))

Syntax hints:   → wi 4   = wceq 1398   ∖ cdif 3211   ∪ cun 3212   ∩ cin 3213  ∅c0 3512  {csn 3694  Ord word 4488  suc csuc 4491

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-ext 2216  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-sn 3700  df-suc 4497

This theorem is referenced by:  phplem3  7121  phplem4  7122  phplem4dom  7129  phplem4on  7135  dif1en  7149