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Mirrors > Home > MPE Home > Th. List > cplem1 | Structured version Visualization version GIF version |
Description: Lemma for the Collection Principle cp 9824. (Contributed by NM, 17-Oct-2003.) |
Ref | Expression |
---|---|
cplem1.1 | ⊢ 𝐶 = {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} |
cplem1.2 | ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 |
Ref | Expression |
---|---|
cplem1 | ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scott0 9819 | . . . . . 6 ⊢ (𝐵 = ∅ ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) | |
2 | cplem1.1 | . . . . . . 7 ⊢ 𝐶 = {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} | |
3 | 2 | eqeq1i 2741 | . . . . . 6 ⊢ (𝐶 = ∅ ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) |
4 | 1, 3 | bitr4i 277 | . . . . 5 ⊢ (𝐵 = ∅ ↔ 𝐶 = ∅) |
5 | 4 | necon3bii 2995 | . . . 4 ⊢ (𝐵 ≠ ∅ ↔ 𝐶 ≠ ∅) |
6 | n0 4305 | . . . 4 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶) | |
7 | 5, 6 | bitri 274 | . . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶) |
8 | 2 | ssrab3 4039 | . . . . . . . 8 ⊢ 𝐶 ⊆ 𝐵 |
9 | 8 | sseli 3939 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐵) |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐵)) |
11 | ssiun2 5006 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) | |
12 | cplem1.2 | . . . . . . . 8 ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 | |
13 | 11, 12 | sseqtrrdi 3994 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ⊆ 𝐷) |
14 | 13 | sseld 3942 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐷)) |
15 | 10, 14 | jcad 513 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → (𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷))) |
16 | inelcm 4423 | . . . . 5 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → (𝐵 ∩ 𝐷) ≠ ∅) | |
17 | 15, 16 | syl6 35 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → (𝐵 ∩ 𝐷) ≠ ∅)) |
18 | 17 | exlimdv 1936 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑤 𝑤 ∈ 𝐶 → (𝐵 ∩ 𝐷) ≠ ∅)) |
19 | 7, 18 | biimtrid 241 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅)) |
20 | 19 | rgen 3065 | 1 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2942 ∀wral 3063 {crab 3406 ∩ cin 3908 ⊆ wss 3909 ∅c0 4281 ∪ ciun 4953 ‘cfv 6494 rankcrnk 9696 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-om 7800 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-r1 9697 df-rank 9698 |
This theorem is referenced by: cplem2 9823 |
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