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Theorem fnessref 34121
 Description: A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Hypotheses
Ref Expression
fnessref.1 𝑋 = 𝐴
fnessref.2 𝑌 = 𝐵
Assertion
Ref Expression
fnessref (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐   𝑋,𝑐   𝑌,𝑐

Proof of Theorem fnessref
Dummy variables 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnerel 34102 . . . . . . 7 Rel Fne
21brrelex2i 5582 . . . . . 6 (𝐴Fne𝐵𝐵 ∈ V)
32adantl 485 . . . . 5 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝐵 ∈ V)
4 rabexg 5204 . . . . 5 (𝐵 ∈ V → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V)
53, 4syl 17 . . . 4 ((𝑋 = 𝑌𝐴Fne𝐵) → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V)
6 ssrab2 3986 . . . . . 6 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵
76a1i 11 . . . . 5 ((𝑋 = 𝑌𝐴Fne𝐵) → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵)
8 fnessref.1 . . . . . . . . . . . 12 𝑋 = 𝐴
98eleq2i 2843 . . . . . . . . . . 11 (𝑡𝑋𝑡 𝐴)
10 eluni 4804 . . . . . . . . . . 11 (𝑡 𝐴 ↔ ∃𝑧(𝑡𝑧𝑧𝐴))
119, 10bitri 278 . . . . . . . . . 10 (𝑡𝑋 ↔ ∃𝑧(𝑡𝑧𝑧𝐴))
12 fnessex 34110 . . . . . . . . . . . . . . . . 17 ((𝐴Fne𝐵𝑧𝐴𝑡𝑧) → ∃𝑥𝐵 (𝑡𝑥𝑥𝑧))
13123expia 1118 . . . . . . . . . . . . . . . 16 ((𝐴Fne𝐵𝑧𝐴) → (𝑡𝑧 → ∃𝑥𝐵 (𝑡𝑥𝑥𝑧)))
1413adantll 713 . . . . . . . . . . . . . . 15 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → (𝑡𝑧 → ∃𝑥𝐵 (𝑡𝑥𝑥𝑧)))
15 sseq2 3920 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
1615rspcev 3543 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝐴𝑥𝑧) → ∃𝑦𝐴 𝑥𝑦)
1716ex 416 . . . . . . . . . . . . . . . . . 18 (𝑧𝐴 → (𝑥𝑧 → ∃𝑦𝐴 𝑥𝑦))
1817adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → (𝑥𝑧 → ∃𝑦𝐴 𝑥𝑦))
1918anim2d 614 . . . . . . . . . . . . . . . 16 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → ((𝑡𝑥𝑥𝑧) → (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2019reximdv 3197 . . . . . . . . . . . . . . 15 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → (∃𝑥𝐵 (𝑡𝑥𝑥𝑧) → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2114, 20syld 47 . . . . . . . . . . . . . 14 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → (𝑡𝑧 → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2221ex 416 . . . . . . . . . . . . 13 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑧𝐴 → (𝑡𝑧 → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦))))
2322com23 86 . . . . . . . . . . . 12 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑡𝑧 → (𝑧𝐴 → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦))))
2423impd 414 . . . . . . . . . . 11 ((𝑋 = 𝑌𝐴Fne𝐵) → ((𝑡𝑧𝑧𝐴) → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2524exlimdv 1934 . . . . . . . . . 10 ((𝑋 = 𝑌𝐴Fne𝐵) → (∃𝑧(𝑡𝑧𝑧𝐴) → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2611, 25syl5bi 245 . . . . . . . . 9 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑡𝑋 → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
27 elunirab 4817 . . . . . . . . 9 (𝑡 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦))
2826, 27syl6ibr 255 . . . . . . . 8 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑡𝑋𝑡 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}))
2928ssrdv 3900 . . . . . . 7 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝑋 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦})
306unissi 4810 . . . . . . . 8 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵
31 simpl 486 . . . . . . . . 9 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝑋 = 𝑌)
32 fnessref.2 . . . . . . . . 9 𝑌 = 𝐵
3331, 32eqtr2di 2810 . . . . . . . 8 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝐵 = 𝑋)
3430, 33sseqtrid 3946 . . . . . . 7 ((𝑋 = 𝑌𝐴Fne𝐵) → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝑋)
3529, 34eqssd 3911 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦})
36 fnessex 34110 . . . . . . . . . 10 ((𝐴Fne𝐵𝑧𝐴𝑡𝑧) → ∃𝑤𝐵 (𝑡𝑤𝑤𝑧))
37363expb 1117 . . . . . . . . 9 ((𝐴Fne𝐵 ∧ (𝑧𝐴𝑡𝑧)) → ∃𝑤𝐵 (𝑡𝑤𝑤𝑧))
3837adantll 713 . . . . . . . 8 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ∃𝑤𝐵 (𝑡𝑤𝑤𝑧))
39 simpl 486 . . . . . . . . . . . . 13 ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → 𝑤𝐵)
4039a1i 11 . . . . . . . . . . . 12 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → 𝑤𝐵))
41 sseq2 3920 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑤𝑦𝑤𝑧))
4241rspcev 3543 . . . . . . . . . . . . . . . 16 ((𝑧𝐴𝑤𝑧) → ∃𝑦𝐴 𝑤𝑦)
4342expcom 417 . . . . . . . . . . . . . . 15 (𝑤𝑧 → (𝑧𝐴 → ∃𝑦𝐴 𝑤𝑦))
4443ad2antll 728 . . . . . . . . . . . . . 14 ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑧𝐴 → ∃𝑦𝐴 𝑤𝑦))
4544com12 32 . . . . . . . . . . . . 13 (𝑧𝐴 → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → ∃𝑦𝐴 𝑤𝑦))
4645ad2antrl 727 . . . . . . . . . . . 12 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → ∃𝑦𝐴 𝑤𝑦))
4740, 46jcad 516 . . . . . . . . . . 11 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑤𝐵 ∧ ∃𝑦𝐴 𝑤𝑦)))
48 sseq1 3919 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑥𝑦𝑤𝑦))
4948rexbidv 3221 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦𝐴 𝑤𝑦))
5049elrab 3604 . . . . . . . . . . 11 (𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ (𝑤𝐵 ∧ ∃𝑦𝐴 𝑤𝑦))
5147, 50syl6ibr 255 . . . . . . . . . 10 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → 𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}))
52 simpr 488 . . . . . . . . . . 11 ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑡𝑤𝑤𝑧))
5352a1i 11 . . . . . . . . . 10 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑡𝑤𝑤𝑧)))
5451, 53jcad 516 . . . . . . . . 9 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ (𝑡𝑤𝑤𝑧))))
5554reximdv2 3195 . . . . . . . 8 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → (∃𝑤𝐵 (𝑡𝑤𝑤𝑧) → ∃𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧)))
5638, 55mpd 15 . . . . . . 7 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ∃𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧))
5756ralrimivva 3120 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → ∀𝑧𝐴𝑡𝑧𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧))
58 eqid 2758 . . . . . . . 8 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}
598, 58isfne2 34106 . . . . . . 7 ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V → (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ (𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ ∀𝑧𝐴𝑡𝑧𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧))))
603, 4, 593syl 18 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ (𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ ∀𝑧𝐴𝑡𝑧𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧))))
6135, 57, 60mpbir2and 712 . . . . 5 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦})
62 sseq1 3919 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝑦𝑧𝑦))
6362rexbidv 3221 . . . . . . . . 9 (𝑥 = 𝑧 → (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦𝐴 𝑧𝑦))
6463elrab 3604 . . . . . . . 8 (𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ (𝑧𝐵 ∧ ∃𝑦𝐴 𝑧𝑦))
65 sseq2 3920 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑧𝑦𝑧𝑤))
6665cbvrexvw 3362 . . . . . . . . . . 11 (∃𝑦𝐴 𝑧𝑦 ↔ ∃𝑤𝐴 𝑧𝑤)
6766biimpi 219 . . . . . . . . . 10 (∃𝑦𝐴 𝑧𝑦 → ∃𝑤𝐴 𝑧𝑤)
6867adantl 485 . . . . . . . . 9 ((𝑧𝐵 ∧ ∃𝑦𝐴 𝑧𝑦) → ∃𝑤𝐴 𝑧𝑤)
6968a1i 11 . . . . . . . 8 ((𝑋 = 𝑌𝐴Fne𝐵) → ((𝑧𝐵 ∧ ∃𝑦𝐴 𝑧𝑦) → ∃𝑤𝐴 𝑧𝑤))
7064, 69syl5bi 245 . . . . . . 7 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → ∃𝑤𝐴 𝑧𝑤))
7170ralrimiv 3112 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → ∀𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}∃𝑤𝐴 𝑧𝑤)
7258, 8isref 22214 . . . . . . 7 ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V → ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴 ↔ (𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ ∀𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}∃𝑤𝐴 𝑧𝑤)))
733, 4, 723syl 18 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴 ↔ (𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ ∀𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}∃𝑤𝐴 𝑧𝑤)))
7435, 71, 73mpbir2and 712 . . . . 5 ((𝑋 = 𝑌𝐴Fne𝐵) → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴)
757, 61, 74jca32 519 . . . 4 ((𝑋 = 𝑌𝐴Fne𝐵) → ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴)))
76 sseq1 3919 . . . . . 6 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → (𝑐𝐵 ↔ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵))
77 breq2 5039 . . . . . . 7 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → (𝐴Fne𝑐𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}))
78 breq1 5038 . . . . . . 7 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → (𝑐Ref𝐴 ↔ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴))
7977, 78anbi12d 633 . . . . . 6 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → ((𝐴Fne𝑐𝑐Ref𝐴) ↔ (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴)))
8076, 79anbi12d 633 . . . . 5 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → ((𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) ↔ ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴))))
8180spcegv 3517 . . . 4 ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V → (({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴)) → ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
825, 75, 81sylc 65 . . 3 ((𝑋 = 𝑌𝐴Fne𝐵) → ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴)))
8382ex 416 . 2 (𝑋 = 𝑌 → (𝐴Fne𝐵 → ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
84 simprrl 780 . . . . 5 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐴Fne𝑐)
85 eqid 2758 . . . . . . . . . . . 12 𝑐 = 𝑐
868, 85fnebas 34108 . . . . . . . . . . 11 (𝐴Fne𝑐𝑋 = 𝑐)
8784, 86syl 17 . . . . . . . . . 10 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑐)
88 simpl 486 . . . . . . . . . 10 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑌)
8987, 88eqtr3d 2795 . . . . . . . . 9 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐 = 𝑌)
9089, 32eqtrdi 2809 . . . . . . . 8 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐 = 𝐵)
91 vuniex 7468 . . . . . . . 8 𝑐 ∈ V
9290, 91eqeltrrdi 2861 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵 ∈ V)
93 uniexb 7490 . . . . . . 7 (𝐵 ∈ V ↔ 𝐵 ∈ V)
9492, 93sylibr 237 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵 ∈ V)
95 simprl 770 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐𝐵)
9685, 32fness 34113 . . . . . 6 ((𝐵 ∈ V ∧ 𝑐𝐵 𝑐 = 𝑌) → 𝑐Fne𝐵)
9794, 95, 89, 96syl3anc 1368 . . . . 5 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐Fne𝐵)
98 fnetr 34115 . . . . 5 ((𝐴Fne𝑐𝑐Fne𝐵) → 𝐴Fne𝐵)
9984, 97, 98syl2anc 587 . . . 4 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐴Fne𝐵)
10099ex 416 . . 3 (𝑋 = 𝑌 → ((𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐴Fne𝐵))
101100exlimdv 1934 . 2 (𝑋 = 𝑌 → (∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐴Fne𝐵))
10283, 101impbid 215 1 (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3860  ∪ cuni 4801   class class class wbr 5035  Refcref 22207  Fnecfne 34100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6298  df-fun 6341  df-fv 6347  df-topgen 16780  df-ref 22210  df-fne 34101 This theorem is referenced by: (None)
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