Theorem phplem3g 7085

Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 7083 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)

Assertion

Ref	Expression
phplem3g	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))

Proof of Theorem phplem3g

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2294	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐴))
2	1	anbi2d 464	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴)))
3		sneq 3684	. . . . . 6 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵})
4	3	difeq2d 3327	. . . . 5 ⊢ (𝑏 = 𝐵 → (suc 𝐴 ∖ {𝑏}) = (suc 𝐴 ∖ {𝐵}))
5	4	breq2d 4105	. . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 ≈ (suc 𝐴 ∖ {𝑏}) ↔ 𝐴 ≈ (suc 𝐴 ∖ {𝐵})))
6	2, 5	imbi12d 234	. . 3 ⊢ (𝑏 = 𝐵 → (((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝑏})) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))))
7		eleq1 2294	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ω ↔ 𝐴 ∈ ω))
8		suceq 4505	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
9	8	eleq2d 2301	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ suc 𝑎 ↔ 𝑏 ∈ suc 𝐴))
10	7, 9	anbi12d 473	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ ω ∧ 𝑏 ∈ suc 𝑎) ↔ (𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴)))
11		id 19	. . . . . . 7 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
12	8	difeq1d 3326	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (suc 𝑎 ∖ {𝑏}) = (suc 𝐴 ∖ {𝑏}))
13	11, 12	breq12d 4106	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ≈ (suc 𝑎 ∖ {𝑏}) ↔ 𝐴 ≈ (suc 𝐴 ∖ {𝑏})))
14	10, 13	imbi12d 234	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎 ∈ ω ∧ 𝑏 ∈ suc 𝑎) → 𝑎 ≈ (suc 𝑎 ∖ {𝑏})) ↔ ((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝑏}))))
15		vex 2806	. . . . . 6 ⊢ 𝑎 ∈ V
16		vex 2806	. . . . . 6 ⊢ 𝑏 ∈ V
17	15, 16	phplem3 7083	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ suc 𝑎) → 𝑎 ≈ (suc 𝑎 ∖ {𝑏}))
18	14, 17	vtoclg 2865	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝑏})))
19	18	anabsi5 581	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝑏}))
20	6, 19	vtoclg 2865	. 2 ⊢ (𝐵 ∈ suc 𝐴 → ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})))
21	20	anabsi7 583	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202   ∖ cdif 3198  {csn 3673   class class class wbr 4093  suc csuc 4468  ωcom 4694   ≈ cen 6950

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-en 6953

This theorem is referenced by:  phplem4dom  7091  phpm  7095  phplem4on  7097