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Mirrors > Home > MPE Home > Th. List > Mathboxes > xppss12 | Structured version Visualization version GIF version |
Description: Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.) |
Ref | Expression |
---|---|
xppss12 | ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssss 4108 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | pssss 4108 | . . 3 ⊢ (𝐶 ⊊ 𝐷 → 𝐶 ⊆ 𝐷) | |
3 | xpss12 5704 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷)) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷)) |
5 | simpl 482 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → 𝐴 ⊊ 𝐵) | |
6 | pssne 4109 | . . . . . 6 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵) | |
7 | 6 | necomd 2994 | . . . . 5 ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ 𝐴) |
8 | neneq 2944 | . . . . . 6 ⊢ (𝐵 ≠ 𝐴 → ¬ 𝐵 = 𝐴) | |
9 | 8 | intnanrd 489 | . . . . 5 ⊢ (𝐵 ≠ 𝐴 → ¬ (𝐵 = 𝐴 ∧ 𝐷 = 𝐶)) |
10 | 5, 7, 9 | 3syl 18 | . . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → ¬ (𝐵 = 𝐴 ∧ 𝐷 = 𝐶)) |
11 | pssn0 42245 | . . . . 5 ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅) | |
12 | pssn0 42245 | . . . . 5 ⊢ (𝐶 ⊊ 𝐷 → 𝐷 ≠ ∅) | |
13 | xp11 6197 | . . . . 5 ⊢ ((𝐵 ≠ ∅ ∧ 𝐷 ≠ ∅) → ((𝐵 × 𝐷) = (𝐴 × 𝐶) ↔ (𝐵 = 𝐴 ∧ 𝐷 = 𝐶))) | |
14 | 11, 12, 13 | syl2an 596 | . . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → ((𝐵 × 𝐷) = (𝐴 × 𝐶) ↔ (𝐵 = 𝐴 ∧ 𝐷 = 𝐶))) |
15 | 10, 14 | mtbird 325 | . . 3 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → ¬ (𝐵 × 𝐷) = (𝐴 × 𝐶)) |
16 | neqne 2946 | . . . 4 ⊢ (¬ (𝐵 × 𝐷) = (𝐴 × 𝐶) → (𝐵 × 𝐷) ≠ (𝐴 × 𝐶)) | |
17 | 16 | necomd 2994 | . . 3 ⊢ (¬ (𝐵 × 𝐷) = (𝐴 × 𝐶) → (𝐴 × 𝐶) ≠ (𝐵 × 𝐷)) |
18 | 15, 17 | syl 17 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → (𝐴 × 𝐶) ≠ (𝐵 × 𝐷)) |
19 | df-pss 3983 | . 2 ⊢ ((𝐴 × 𝐶) ⊊ (𝐵 × 𝐷) ↔ ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐷) ∧ (𝐴 × 𝐶) ≠ (𝐵 × 𝐷))) | |
20 | 4, 18, 19 | sylanbrc 583 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ≠ wne 2938 ⊆ wss 3963 ⊊ wpss 3964 ∅c0 4339 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: (None) |
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