Theorem pr2neOLD 9997

Description: Obsolete version of pr2ne 9996 as of 30-Dec-2024. (Contributed by FL, 14-Feb-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pr2neOLD	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵))

Proof of Theorem pr2neOLD

Step	Hyp	Ref	Expression
1		preq2 4738	. . . . 5 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴})
2	1	eqcoms 2741	. . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴, 𝐴})
3		enpr1g 9017	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐶 → {𝐴, 𝐴} ≈ 1o)
4		entr 8999	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝐵} ≈ {𝐴, 𝐴} ∧ {𝐴, 𝐴} ≈ 1o) → {𝐴, 𝐵} ≈ 1o)
5		1sdom2 9237	. . . . . . . . . . . . . . 15 ⊢ 1o ≺ 2o
6		sdomnen 8974	. . . . . . . . . . . . . . 15 ⊢ (1o ≺ 2o → ¬ 1o ≈ 2o)
7	5, 6	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ¬ 1o ≈ 2o
8		ensym 8996	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ≈ 1o → 1o ≈ {𝐴, 𝐵})
9		entr 8999	. . . . . . . . . . . . . . . 16 ⊢ ((1o ≈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ≈ 2o) → 1o ≈ 2o)
10	9	ex 414	. . . . . . . . . . . . . . 15 ⊢ (1o ≈ {𝐴, 𝐵} → ({𝐴, 𝐵} ≈ 2o → 1o ≈ 2o))
11	8, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ≈ 1o → ({𝐴, 𝐵} ≈ 2o → 1o ≈ 2o))
12	7, 11	mtoi 198	. . . . . . . . . . . . 13 ⊢ ({𝐴, 𝐵} ≈ 1o → ¬ {𝐴, 𝐵} ≈ 2o)
13	12	a1d 25	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐵} ≈ 1o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ¬ {𝐴, 𝐵} ≈ 2o))
14	4, 13	syl 17	. . . . . . . . . . 11 ⊢ (({𝐴, 𝐵} ≈ {𝐴, 𝐴} ∧ {𝐴, 𝐴} ≈ 1o) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ¬ {𝐴, 𝐵} ≈ 2o))
15	14	ex 414	. . . . . . . . . 10 ⊢ ({𝐴, 𝐵} ≈ {𝐴, 𝐴} → ({𝐴, 𝐴} ≈ 1o → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ¬ {𝐴, 𝐵} ≈ 2o)))
16		prex 5432	. . . . . . . . . . 11 ⊢ {𝐴, 𝐵} ∈ V
17		eqeng 8979	. . . . . . . . . . 11 ⊢ ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} = {𝐴, 𝐴} → {𝐴, 𝐵} ≈ {𝐴, 𝐴}))
18	16, 17	ax-mp 5	. . . . . . . . . 10 ⊢ ({𝐴, 𝐵} = {𝐴, 𝐴} → {𝐴, 𝐵} ≈ {𝐴, 𝐴})
19	15, 18	syl11 33	. . . . . . . . 9 ⊢ ({𝐴, 𝐴} ≈ 1o → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ¬ {𝐴, 𝐵} ≈ 2o)))
20	19	a1dd 50	. . . . . . . 8 ⊢ ({𝐴, 𝐴} ≈ 1o → ({𝐴, 𝐵} = {𝐴, 𝐴} → (𝐵 ∈ 𝐷 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ¬ {𝐴, 𝐵} ≈ 2o))))
21	3, 20	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → ({𝐴, 𝐵} = {𝐴, 𝐴} → (𝐵 ∈ 𝐷 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ¬ {𝐴, 𝐵} ≈ 2o))))
22	21	com23 86	. . . . . 6 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ¬ {𝐴, 𝐵} ≈ 2o))))
23	22	imp 408	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐴, 𝐴} → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ¬ {𝐴, 𝐵} ≈ 2o)))
24	23	pm2.43a 54	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐴, 𝐴} → ¬ {𝐴, 𝐵} ≈ 2o))
25	2, 24	syl5 34	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2o))
26	25	necon2ad 2956	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o → 𝐴 ≠ 𝐵))
27		enpr2 9994	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o)
28	27	3expia 1122	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2o))
29	26, 28	impbid 211	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475  {cpr 4630   class class class wbr 5148  1_oc1o 8456  2_oc2o 8457   ≈ cen 8933   ≺ csdm 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-1o 8463  df-2o 8464  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939

This theorem is referenced by: (None)