Theorem difopab 4680

Description: The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
difopab	⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem difopab

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 4674	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		reldif 4667	. . 3 ⊢ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} → Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
3	1, 2	ax-mp 5	. 2 ⊢ Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4		relopab 4674	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
5		sbcan 2955	. . . 4 ⊢ ([𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
6		sbcan 2955	. . . . 5 ⊢ ([𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ ([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓))
7	6	sbcbii 2972	. . . 4 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ [𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓))
8		opelopabsb 4190	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9		vex 2692	. . . . . . 7 ⊢ 𝑧 ∈ V
10		sbcng 2953	. . . . . . 7 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓))
11	9, 10	ax-mp 5	. . . . . 6 ⊢ ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
12		vex 2692	. . . . . . . 8 ⊢ 𝑤 ∈ V
13		sbcng 2953	. . . . . . . 8 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓))
14	12, 13	ax-mp 5	. . . . . . 7 ⊢ ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓)
15	14	sbcbii 2972	. . . . . 6 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓)
16		opelopabsb 4190	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
17	16	notbii 658	. . . . . 6 ⊢ (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
18	11, 15, 17	3bitr4ri 212	. . . . 5 ⊢ (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓)
19	8, 18	anbi12i 456	. . . 4 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
20	5, 7, 19	3bitr4ri 212	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
21		eldif 3085	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
22		opelopabsb 4190	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)} ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
23	20, 21, 22	3bitr4i 211	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)})
24	3, 4, 23	eqrelriiv 4641	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  Vcvv 2689  [wsbc 2913   ∖ cdif 3073  ⟨cop 3535  {copab 3996  Rel wrel 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-xp 4553  df-rel 4554

This theorem is referenced by: (None)