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Theorem nodenselem7 31086
 Description: Lemma for nodense 31088. 𝐴 and 𝐵 are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
Assertion
Ref Expression
nodenselem7 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎   𝐶,𝑎

Proof of Theorem nodenselem7
StepHypRef Expression
1 nodenselem4 31083 . . . . 5 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
21adantrl 748 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On)
3 onelon 5665 . . . . 5 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → 𝐶 ∈ On)
43ex 449 . . . 4 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → 𝐶 ∈ On))
52, 4syl 17 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → 𝐶 ∈ On))
62, 3sylan 487 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → 𝐶 ∈ On)
7 ontri1 5674 . . . . . . . . . . . . . 14 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝐶 ∈ On) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶 ↔ ¬ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
87con2bid 343 . . . . . . . . . . . . 13 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝐶 ∈ On) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶))
98biimpd 218 . . . . . . . . . . . 12 (( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On ∧ 𝐶 ∈ On) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶))
109ex 449 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On → (𝐶 ∈ On → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)))
1110com23 84 . . . . . . . . . 10 ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ On → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐶 ∈ On → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)))
122, 11syl 17 . . . . . . . . 9 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐶 ∈ On → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)))
1312imp 444 . . . . . . . 8 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → (𝐶 ∈ On → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶))
146, 13mpd 15 . . . . . . 7 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ¬ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)
15 intss1 4427 . . . . . . 7 (𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ⊆ 𝐶)
1614, 15nsyl 134 . . . . . 6 ((((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) ∧ 𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ¬ 𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})
1716ex 449 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ 𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
18 fveq2 6103 . . . . . . . 8 (𝑎 = 𝐶 → (𝐴𝑎) = (𝐴𝐶))
19 fveq2 6103 . . . . . . . 8 (𝑎 = 𝐶 → (𝐵𝑎) = (𝐵𝐶))
2018, 19neeq12d 2843 . . . . . . 7 (𝑎 = 𝐶 → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴𝐶) ≠ (𝐵𝐶)))
2120elrab 3331 . . . . . 6 (𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ (𝐶 ∈ On ∧ (𝐴𝐶) ≠ (𝐵𝐶)))
2221notbii 309 . . . . 5 𝐶 ∈ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ↔ ¬ (𝐶 ∈ On ∧ (𝐴𝐶) ≠ (𝐵𝐶)))
2317, 22syl6ib 240 . . . 4 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ (𝐶 ∈ On ∧ (𝐴𝐶) ≠ (𝐵𝐶))))
24 imnan 437 . . . 4 ((𝐶 ∈ On → ¬ (𝐴𝐶) ≠ (𝐵𝐶)) ↔ ¬ (𝐶 ∈ On ∧ (𝐴𝐶) ≠ (𝐵𝐶)))
2523, 24syl6ibr 241 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐶 ∈ On → ¬ (𝐴𝐶) ≠ (𝐵𝐶))))
265, 25mpdd 42 . 2 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → ¬ (𝐴𝐶) ≠ (𝐵𝐶)))
27 df-ne 2782 . . 3 ((𝐴𝐶) ≠ (𝐵𝐶) ↔ ¬ (𝐴𝐶) = (𝐵𝐶))
2827con2bii 346 . 2 ((𝐴𝐶) = (𝐵𝐶) ↔ ¬ (𝐴𝐶) ≠ (𝐵𝐶))
2926, 28syl6ibr 241 1 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → (𝐶 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} → (𝐴𝐶) = (𝐵𝐶)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ⊆ wss 3540  ∩ cint 4410   class class class wbr 4583  Oncon0 5640  ‘cfv 5804   No csur 31037
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