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Theorem nodenselem8 27737
Description: Lemma for nodense 27738. Give a condition for surreal less-than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)
Assertion
Ref Expression
nodenselem8 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 ↔ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem8
StepHypRef Expression
1 nodenselem5 27734 . . . . 5 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))
21exp32 420 . . . 4 ((𝐴 No 𝐵 No ) → (( bday 𝐴) = ( bday 𝐵) → (𝐴 <s 𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴))))
323impia 1117 . . 3 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)))
4 sltval2 27702 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
543adant3 1132 . . . 4 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))
6 fvex 6918 . . . . . 6 (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
7 fvex 6918 . . . . . 6 (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ V
86, 7brtp 5527 . . . . 5 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ↔ (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
9 eleq2 2829 . . . . . . . . . . . . 13 (( bday 𝐴) = ( bday 𝐵) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) ↔ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵)))
109biimpd 229 . . . . . . . . . . . 12 (( bday 𝐴) = ( bday 𝐵) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵)))
11 nosgnn0 27704 . . . . . . . . . . . . . . 15 ¬ ∅ ∈ {1o, 2o}
12 nofnbday 27698 . . . . . . . . . . . . . . . . 17 (𝐵 No 𝐵 Fn ( bday 𝐵))
13 fnfvelrn 7099 . . . . . . . . . . . . . . . . . 18 ((𝐵 Fn ( bday 𝐵) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵)) → (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ ran 𝐵)
14 eleq1 2828 . . . . . . . . . . . . . . . . . 18 ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ ran 𝐵 ↔ ∅ ∈ ran 𝐵))
1513, 14syl5ibcom 245 . . . . . . . . . . . . . . . . 17 ((𝐵 Fn ( bday 𝐵) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵)) → ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ∅ ∈ ran 𝐵))
1612, 15sylan 580 . . . . . . . . . . . . . . . 16 ((𝐵 No {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵)) → ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ∅ ∈ ran 𝐵))
17 norn 27697 . . . . . . . . . . . . . . . . . 18 (𝐵 No → ran 𝐵 ⊆ {1o, 2o})
1817sseld 3981 . . . . . . . . . . . . . . . . 17 (𝐵 No → (∅ ∈ ran 𝐵 → ∅ ∈ {1o, 2o}))
1918adantr 480 . . . . . . . . . . . . . . . 16 ((𝐵 No {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵)) → (∅ ∈ ran 𝐵 → ∅ ∈ {1o, 2o}))
2016, 19syld 47 . . . . . . . . . . . . . . 15 ((𝐵 No {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵)) → ((𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ∅ ∈ {1o, 2o}))
2111, 20mtoi 199 . . . . . . . . . . . . . 14 ((𝐵 No {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵)) → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
2221ex 412 . . . . . . . . . . . . 13 (𝐵 No → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵) → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅))
2322adantl 481 . . . . . . . . . . . 12 ((𝐴 No 𝐵 No ) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐵) → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅))
2410, 23syl9r 78 . . . . . . . . . . 11 ((𝐴 No 𝐵 No ) → (( bday 𝐴) = ( bday 𝐵) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)))
25243impia 1117 . . . . . . . . . 10 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅))
2625imp 406 . . . . . . . . 9 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ¬ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
2726intnand 488 . . . . . . . 8 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ¬ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅))
28 nofnbday 27698 . . . . . . . . . . . . 13 (𝐴 No 𝐴 Fn ( bday 𝐴))
29 fnfvelrn 7099 . . . . . . . . . . . . . 14 ((𝐴 Fn ( bday 𝐴) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ ran 𝐴)
30 eleq1 2828 . . . . . . . . . . . . . 14 ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) ∈ ran 𝐴 ↔ ∅ ∈ ran 𝐴))
3129, 30syl5ibcom 245 . . . . . . . . . . . . 13 ((𝐴 Fn ( bday 𝐴) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ∅ ∈ ran 𝐴))
3228, 31sylan 580 . . . . . . . . . . . 12 ((𝐴 No {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ∅ ∈ ran 𝐴))
33 norn 27697 . . . . . . . . . . . . . 14 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
3433sseld 3981 . . . . . . . . . . . . 13 (𝐴 No → (∅ ∈ ran 𝐴 → ∅ ∈ {1o, 2o}))
3534adantr 480 . . . . . . . . . . . 12 ((𝐴 No {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → (∅ ∈ ran 𝐴 → ∅ ∈ {1o, 2o}))
3632, 35syld 47 . . . . . . . . . . 11 ((𝐴 No {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ → ∅ ∈ {1o, 2o}))
3711, 36mtoi 199 . . . . . . . . . 10 ((𝐴 No {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ¬ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
38373ad2antl1 1185 . . . . . . . . 9 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ¬ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅)
3938intnanrd 489 . . . . . . . 8 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ¬ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o))
40 3orel13 1488 . . . . . . . 8 ((¬ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∧ ¬ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)) → ((((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
4127, 39, 40syl2anc 584 . . . . . . 7 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴)) → ((((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
4241ex 412 . . . . . 6 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → ((((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o))))
4342com23 86 . . . . 5 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → ((((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o))))
448, 43biimtrid 242 . . . 4 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o))))
455, 44sylbid 240 . . 3 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 → ( {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ ( bday 𝐴) → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o))))
463, 45mpdd 43 . 2 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 → ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
47 3mix2 1331 . . . 4 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) → (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) ∨ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = ∅ ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
4847, 8sylibr 234 . . 3 (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) → (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}))
4948, 5imbitrrid 246 . 2 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o) → 𝐴 <s 𝐵))
5046, 49impbid 212 1 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 ↔ ((𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 1o ∧ (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}) = 2o)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3o 1085  w3a 1086   = wceq 1539  wcel 2107  wne 2939  {crab 3435  c0 4332  {cpr 4627  {ctp 4629  cop 4631   cint 4945   class class class wbr 5142  ran crn 5685  Oncon0 6383   Fn wfn 6555  cfv 6560  1oc1o 8500  2oc2o 8501   No csur 27685   <s cslt 27686   bday cbday 27687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689  df-bday 27690
This theorem is referenced by:  nodense  27738
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