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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjss | Structured version Visualization version GIF version |
Description: Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) |
Ref | Expression |
---|---|
disjss | ⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 5715 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | funALTVss 36433 | . . . 4 ⊢ (◡𝐴 ⊆ ◡𝐵 → ( FunALTV ◡𝐵 → FunALTV ◡𝐴)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV ◡𝐵 → FunALTV ◡𝐴)) |
4 | relss 5627 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) | |
5 | 3, 4 | anim12d 612 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (( FunALTV ◡𝐵 ∧ Rel 𝐵) → ( FunALTV ◡𝐴 ∧ Rel 𝐴))) |
6 | dfdisjALTV 36447 | . 2 ⊢ ( Disj 𝐵 ↔ ( FunALTV ◡𝐵 ∧ Rel 𝐵)) | |
7 | dfdisjALTV 36447 | . 2 ⊢ ( Disj 𝐴 ↔ ( FunALTV ◡𝐴 ∧ Rel 𝐴)) | |
8 | 5, 6, 7 | 3imtr4g 299 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ⊆ wss 3843 ◡ccnv 5524 Rel wrel 5530 FunALTV wfunALTV 35987 Disj wdisjALTV 35990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-coss 36160 df-cnvrefrel 36266 df-funALTV 36416 df-disjALTV 36439 |
This theorem is referenced by: disjssi 36466 disjssd 36467 |
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