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| Mirrors > Home > MPE Home > Th. List > psseq2 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.) |
| Ref | Expression |
|---|---|
| psseq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseq2 4010 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) | |
| 2 | neeq2 3004 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) | |
| 3 | 1, 2 | anbi12d 632 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ 𝐴) ↔ (𝐶 ⊆ 𝐵 ∧ 𝐶 ≠ 𝐵))) |
| 4 | df-pss 3971 | . 2 ⊢ (𝐶 ⊊ 𝐴 ↔ (𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ 𝐴)) | |
| 5 | df-pss 3971 | . 2 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ 𝐶 ≠ 𝐵)) | |
| 6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ≠ wne 2940 ⊆ wss 3951 ⊊ wpss 3952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ne 2941 df-ss 3968 df-pss 3971 |
| This theorem is referenced by: psseq2i 4093 psseq2d 4096 psssstr 4109 brrpssg 7745 sorpssint 7753 pssnn 9208 php 9247 phpOLD 9259 php2OLD 9260 isfin4 10337 fin2i2 10358 elnp 11027 elnpi 11028 ltprord 11070 pgpfac1lem1 20094 pgpfac1lem5 20099 lbsextlem4 21163 alexsubALTlem4 24058 spansncv 31672 cvbr 32301 cvcon3 32303 cvnbtwn 32305 cvbr4i 32386 ssdifidlprm 33486 ssmxidl 33502 dfon2lem6 35789 dfon2lem7 35790 dfon2lem8 35791 dfon2 35793 lcvbr 39022 lcvnbtwn 39026 lsatcv0 39032 lsat0cv 39034 islshpcv 39054 mapdcv 41662 pssn0 42266 |
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