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Mirrors > Home > MPE Home > Th. List > kmlem5 | Structured version Visualization version GIF version |
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
Ref | Expression |
---|---|
kmlem5 | ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3737 | . . . 4 ⊢ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤})) ⊆ 𝑤 | |
2 | sslin 3839 | . . . 4 ⊢ ((𝑤 ∖ ∪ (𝑥 ∖ {𝑤})) ⊆ 𝑤 → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) |
4 | kmlem4 8975 | . . 3 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅) | |
5 | 3, 4 | syl5sseq 3653 | . 2 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ∅) |
6 | ss0b 3973 | . 2 ⊢ (((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ∅ ↔ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅) | |
7 | 5, 6 | sylib 208 | 1 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ≠ wne 2794 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 ∪ cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-uni 4437 |
This theorem is referenced by: kmlem9 8980 |
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